http://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_4&feed=atom&action=historyCRN 11378: HW 4 - Revision history2024-03-28T12:40:05ZRevision history for this page on the wikiMediaWiki 1.19.1http://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_4&diff=2828&oldid=prevHelmutKnaust at 18:14, 12 November 20192019-11-12T18:14:00Z<p></p>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">25</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">16</ins>.'''  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let $f</del>:<del class="diffchange diffchange-inline">[a,b]\to\mathbb{R}</del>$ <del class="diffchange diffchange-inline">be </del>an <del class="diffchange diffchange-inline">increasing function. Show that </del>$\<del class="diffchange diffchange-inline">lim_{</del>x<del class="diffchange diffchange-inline">\to a}f(</del>x<del class="diffchange diffchange-inline">)</del>$ <del class="diffchange diffchange-inline">exists. What can you say about the relationship between this limit and </del>$<del class="diffchange diffchange-inline">f</del>(<del class="diffchange diffchange-inline">a</del>)<del class="diffchange diffchange-inline">$</del>?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Show</ins>: <ins class="diffchange diffchange-inline">If </ins>$<ins class="diffchange diffchange-inline">x$ is </ins>an <ins class="diffchange diffchange-inline">accumulation point of </ins>$<ins class="diffchange diffchange-inline">A</ins>\<ins class="diffchange diffchange-inline">cup B$, then $</ins>x<ins class="diffchange diffchange-inline">$ is an accumulation point of $A$, or $</ins>x$ <ins class="diffchange diffchange-inline">is an accumulation point of $B</ins>$ (<ins class="diffchange diffchange-inline">or both</ins>)<ins class="diffchange diffchange-inline">. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Does the result also hold for a countably infinite collection of sets</ins>? <ins class="diffchange diffchange-inline">Give a proof, or provide a counterexample.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">26</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">17</ins>.''' <ins class="diffchange diffchange-inline">Prove: A subset </ins>$<ins class="diffchange diffchange-inline">F</ins>\<ins class="diffchange diffchange-inline">subseteq </ins>\mathbb{R}$ <ins class="diffchange diffchange-inline">is closed if and only if every Cauchy sequence contained in </ins>$<ins class="diffchange diffchange-inline">F</ins>$ <ins class="diffchange diffchange-inline">converges to an element </ins>in $<ins class="diffchange diffchange-inline">F</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let </del>$<del class="diffchange diffchange-inline">f,g:</del>\<del class="diffchange diffchange-inline">mathbb{R}\to</del>\mathbb{R}$ <del class="diffchange diffchange-inline">be two continuous functions. Define </del>$<del class="diffchange diffchange-inline">h(x)=\max\{f(x),g(x)\}</del>$ <del class="diffchange diffchange-inline">for all $x\</del>in<del class="diffchange diffchange-inline">\mathbb{R}</del>$<del class="diffchange diffchange-inline">. Show that $h$ is continuous on $\mathbb{R}</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">27</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">18</ins>.''' <ins class="diffchange diffchange-inline">Find all accumulation points of the set </ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let $f:</del>\<del class="diffchange diffchange-inline">mathbb</del>{<del class="diffchange diffchange-inline">R</del>}\<del class="diffchange diffchange-inline">to\mathbb</del>{<del class="diffchange diffchange-inline">R</del>}<del class="diffchange diffchange-inline">$ be continuous on $</del>\mathbb{<del class="diffchange diffchange-inline">R</del>}<del class="diffchange diffchange-inline">$, and assume </del>that <del class="diffchange diffchange-inline">for all </del>$\<del class="diffchange diffchange-inline">varepsilon>0$ there is an $N>0$ such that $|f</del>(<del class="diffchange diffchange-inline">x</del>)<del class="diffchange diffchange-inline">|<</del>\<del class="diffchange diffchange-inline">varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $</del>\<del class="diffchange diffchange-inline">mathbb{R}</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\<ins class="diffchange diffchange-inline">[\left\</ins>{<ins class="diffchange diffchange-inline">\frac{1}{m</ins>}<ins class="diffchange diffchange-inline">+</ins>\<ins class="diffchange diffchange-inline">frac</ins>{<ins class="diffchange diffchange-inline">1}{n</ins>}<ins class="diffchange diffchange-inline">\ |\ m,n\in</ins>\mathbb{<ins class="diffchange diffchange-inline">N</ins>}<ins class="diffchange diffchange-inline">\right\}\]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Remember </ins>that $<ins class="diffchange diffchange-inline">A=B</ins>\ <ins class="diffchange diffchange-inline"> \Leftrightarrow\  </ins>(<ins class="diffchange diffchange-inline">A\subseteq B</ins>)\<ins class="diffchange diffchange-inline">wedge (B</ins>\<ins class="diffchange diffchange-inline">subseteq A)</ins>$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">28</del>.'''</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">19</ins>.''' <ins class="diffchange diffchange-inline">Show: If </ins>$<ins class="diffchange diffchange-inline">X</ins>\<ins class="diffchange diffchange-inline">subseteq </ins>\mathbb{R}$ is <ins class="diffchange diffchange-inline">both open and closed, then </ins>$<ins class="diffchange diffchange-inline">X=</ins>\<ins class="diffchange diffchange-inline">mathbb{R}</ins>$ <ins class="diffchange diffchange-inline">or </ins>$<ins class="diffchange diffchange-inline">X=</ins>\<ins class="diffchange diffchange-inline">emptyset</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let </del>$<del class="diffchange diffchange-inline">f:[a,b]</del>\<del class="diffchange diffchange-inline">to</del>\mathbb{R}$ <del class="diffchange diffchange-inline">be a function. We say $f$ satisfies $(*)$ if there </del>is <del class="diffchange diffchange-inline">an </del>$<del class="diffchange diffchange-inline">M>0$ such that $|f(x)-f(y)|\leq M</del>\<del class="diffchange diffchange-inline">cdot |x-y|</del>$ <del class="diffchange diffchange-inline">for all </del>$<del class="diffchange diffchange-inline">x,y</del>\<del class="diffchange diffchange-inline">in [a,b]</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Let $g</del>:[<del class="diffchange diffchange-inline">0</del>,1<del class="diffchange diffchange-inline">]</del>\<del class="diffchange diffchange-inline">to</del>\<del class="diffchange diffchange-inline">mathbb</del>{<del class="diffchange diffchange-inline">R</del>}<del class="diffchange diffchange-inline">$ be given by $g(x)</del>=\<del class="diffchange diffchange-inline">sqrt</del>{<del class="diffchange diffchange-inline">x</del>}<del class="diffchange diffchange-inline">$. Show that $g$ does not satisfy $(*)$.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Problem 20.''' Consider the following sets</ins>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Is $g$ uniformly continuous on $</del>[<del class="diffchange diffchange-inline">a</del>,<del class="diffchange diffchange-inline">b</del>]<del class="diffchange diffchange-inline">$? Is $g$ uniformly continuous on $(a</del>,<del class="diffchange diffchange-inline">b)$? Explain!</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\</ins>[<ins class="diffchange diffchange-inline">A=\left\{1</ins>,<ins class="diffchange diffchange-inline">\frac{</ins>1<ins class="diffchange diffchange-inline">}{2},</ins>\<ins class="diffchange diffchange-inline">frac{1}{3},</ins>\<ins class="diffchange diffchange-inline">frac</ins>{<ins class="diffchange diffchange-inline">1</ins>}<ins class="diffchange diffchange-inline">{4}\ldots\right\},\quad B</ins>=\<ins class="diffchange diffchange-inline">left\</ins>{<ins class="diffchange diffchange-inline">1,\frac{1</ins>}<ins class="diffchange diffchange-inline">{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap</ins>[<ins class="diffchange diffchange-inline">0</ins>,<ins class="diffchange diffchange-inline">1</ins>]<ins class="diffchange diffchange-inline">\]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">For the sets that are compact</ins>, <ins class="diffchange diffchange-inline">explain why. For the other ones, show that they have an open cover without finite subcover.</ins></div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_4&diff=2827&oldid=prevHelmutKnaust at 18:12, 12 November 20192019-11-12T18:12:59Z<p></p>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">16</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">25</ins>.'''  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Show</del>: <del class="diffchange diffchange-inline">If </del>$<del class="diffchange diffchange-inline">x$ is </del>an <del class="diffchange diffchange-inline">accumulation point of </del>$<del class="diffchange diffchange-inline">A</del>\<del class="diffchange diffchange-inline">cup B$, then $</del>x<del class="diffchange diffchange-inline">$ is an accumulation point of $A$, or $</del>x$ <del class="diffchange diffchange-inline">is an accumulation point of $B</del>$ (<del class="diffchange diffchange-inline">or both</del>)<del class="diffchange diffchange-inline">. </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let $f</ins>:<ins class="diffchange diffchange-inline">[a,b]\to\mathbb{R}</ins>$ <ins class="diffchange diffchange-inline">be </ins>an <ins class="diffchange diffchange-inline">increasing function. Show that </ins>$\<ins class="diffchange diffchange-inline">lim_{</ins>x<ins class="diffchange diffchange-inline">\to a}f(</ins>x<ins class="diffchange diffchange-inline">)</ins>$ <ins class="diffchange diffchange-inline">exists. What can you say about the relationship between this limit and </ins>$<ins class="diffchange diffchange-inline">f</ins>(<ins class="diffchange diffchange-inline">a</ins>)<ins class="diffchange diffchange-inline">$</ins>?</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Does the result also hold for a countably infinite collection of sets</del>? <del class="diffchange diffchange-inline">Give a proof, or provide a counterexample.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">17</del>.''' <del class="diffchange diffchange-inline">Prove: A subset </del>$<del class="diffchange diffchange-inline">F</del>\<del class="diffchange diffchange-inline">subseteq </del>\mathbb{R}$ <del class="diffchange diffchange-inline">is closed if and only if every Cauchy sequence contained in </del>$<del class="diffchange diffchange-inline">F</del>$ <del class="diffchange diffchange-inline">converges to an element </del>in $<del class="diffchange diffchange-inline">F</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">26</ins>.'''  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let </ins>$<ins class="diffchange diffchange-inline">f,g:</ins>\<ins class="diffchange diffchange-inline">mathbb{R}\to</ins>\mathbb{R}$ <ins class="diffchange diffchange-inline">be two continuous functions. Define </ins>$<ins class="diffchange diffchange-inline">h(x)=\max\{f(x),g(x)\}</ins>$ <ins class="diffchange diffchange-inline">for all $x\</ins>in<ins class="diffchange diffchange-inline">\mathbb{R}</ins>$<ins class="diffchange diffchange-inline">. Show that $h$ is continuous on $\mathbb{R}</ins>$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">18</del>.''' <del class="diffchange diffchange-inline">Find all accumulation points of the set </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">27</ins>.'''  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\<del class="diffchange diffchange-inline">[\left\</del>{<del class="diffchange diffchange-inline">\frac{1}{m</del>}<del class="diffchange diffchange-inline">+</del>\<del class="diffchange diffchange-inline">frac</del>{<del class="diffchange diffchange-inline">1}{n</del>}<del class="diffchange diffchange-inline">\ |\ m,n\in</del>\mathbb{<del class="diffchange diffchange-inline">N</del>}<del class="diffchange diffchange-inline">\right\}\]</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let $f:</ins>\<ins class="diffchange diffchange-inline">mathbb</ins>{<ins class="diffchange diffchange-inline">R</ins>}\<ins class="diffchange diffchange-inline">to\mathbb</ins>{<ins class="diffchange diffchange-inline">R</ins>}<ins class="diffchange diffchange-inline">$ be continuous on $</ins>\mathbb{<ins class="diffchange diffchange-inline">R</ins>}<ins class="diffchange diffchange-inline">$, and assume </ins>that <ins class="diffchange diffchange-inline">for all </ins>$\<ins class="diffchange diffchange-inline">varepsilon>0$ there is an $N>0$ such that $|f</ins>(<ins class="diffchange diffchange-inline">x</ins>)<ins class="diffchange diffchange-inline">|<</ins>\<ins class="diffchange diffchange-inline">varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $</ins>\<ins class="diffchange diffchange-inline">mathbb{R}</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Remember </del>that $<del class="diffchange diffchange-inline">A=B</del>\ <del class="diffchange diffchange-inline"> \Leftrightarrow\  </del>(<del class="diffchange diffchange-inline">A\subseteq B</del>)\<del class="diffchange diffchange-inline">wedge (B</del>\<del class="diffchange diffchange-inline">subseteq A)</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">19</del>.''' <del class="diffchange diffchange-inline">Show: If </del>$<del class="diffchange diffchange-inline">X</del>\<del class="diffchange diffchange-inline">subseteq </del>\mathbb{R}$ is <del class="diffchange diffchange-inline">both open and closed, then </del>$<del class="diffchange diffchange-inline">X=</del>\<del class="diffchange diffchange-inline">mathbb{R}</del>$ <del class="diffchange diffchange-inline">or </del>$<del class="diffchange diffchange-inline">X=</del>\<del class="diffchange diffchange-inline">emptyset</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">28</ins>.'''</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let </ins>$<ins class="diffchange diffchange-inline">f:[a,b]</ins>\<ins class="diffchange diffchange-inline">to</ins>\mathbb{R}$ <ins class="diffchange diffchange-inline">be a function. We say $f$ satisfies $(*)$ if there </ins>is <ins class="diffchange diffchange-inline">an </ins>$<ins class="diffchange diffchange-inline">M>0$ such that $|f(x)-f(y)|</ins>\<ins class="diffchange diffchange-inline">leq M\cdot |x-y|</ins>$ <ins class="diffchange diffchange-inline">for all </ins>$<ins class="diffchange diffchange-inline">x,y</ins>\<ins class="diffchange diffchange-inline">in [a,b]</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">'''Problem 20.''' Consider the following sets</del>:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Let $g</ins>:[<ins class="diffchange diffchange-inline">0</ins>,1<ins class="diffchange diffchange-inline">]</ins>\<ins class="diffchange diffchange-inline">to</ins>\<ins class="diffchange diffchange-inline">mathbb</ins>{<ins class="diffchange diffchange-inline">R</ins>}<ins class="diffchange diffchange-inline">$ be given by $g(x)</ins>=\<ins class="diffchange diffchange-inline">sqrt</ins>{<ins class="diffchange diffchange-inline">x</ins>}<ins class="diffchange diffchange-inline">$. Show that $g$ does not satisfy $(*)$.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">\</del>[<del class="diffchange diffchange-inline">A=\left\{1</del>,<del class="diffchange diffchange-inline">\frac{</del>1<del class="diffchange diffchange-inline">}{2},</del>\<del class="diffchange diffchange-inline">frac{1}{3},</del>\<del class="diffchange diffchange-inline">frac</del>{<del class="diffchange diffchange-inline">1</del>}<del class="diffchange diffchange-inline">{4}\ldots\right\},\quad B</del>=\<del class="diffchange diffchange-inline">left\</del>{<del class="diffchange diffchange-inline">1,\frac{1</del>}<del class="diffchange diffchange-inline">{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap</del>[<del class="diffchange diffchange-inline">0</del>,<del class="diffchange diffchange-inline">1</del>]<del class="diffchange diffchange-inline">\]</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Is $g$ uniformly continuous on $</ins>[<ins class="diffchange diffchange-inline">a</ins>,<ins class="diffchange diffchange-inline">b</ins>]<ins class="diffchange diffchange-inline">$? Is $g$ uniformly continuous on $(a</ins>,<ins class="diffchange diffchange-inline">b)$? Explain!</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">For the sets that are compact</del>, <del class="diffchange diffchange-inline">explain why. For the other ones, show that they have an open cover without finite subcover.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_4&diff=2826&oldid=prevHelmutKnaust at 18:11, 12 November 20192019-11-12T18:11:40Z<p></p>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">25</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">16</ins>.'''  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let $f</del>:<del class="diffchange diffchange-inline">[a,b]\to\mathbb{R}</del>$ <del class="diffchange diffchange-inline">be </del>an <del class="diffchange diffchange-inline">increasing function. Show that </del>$\<del class="diffchange diffchange-inline">lim_{</del>x<del class="diffchange diffchange-inline">\to a}f(</del>x<del class="diffchange diffchange-inline">)</del>$ <del class="diffchange diffchange-inline">exists. What can you say about the relationship between this limit and </del>$<del class="diffchange diffchange-inline">f</del>(<del class="diffchange diffchange-inline">a</del>)<del class="diffchange diffchange-inline">$</del>?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Show</ins>: <ins class="diffchange diffchange-inline">If </ins>$<ins class="diffchange diffchange-inline">x$ is </ins>an <ins class="diffchange diffchange-inline">accumulation point of </ins>$<ins class="diffchange diffchange-inline">A</ins>\<ins class="diffchange diffchange-inline">cup B$, then $</ins>x<ins class="diffchange diffchange-inline">$ is an accumulation point of $A$, or $</ins>x$ <ins class="diffchange diffchange-inline">is an accumulation point of $B</ins>$ (<ins class="diffchange diffchange-inline">or both</ins>)<ins class="diffchange diffchange-inline">. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Does the result also hold for a countably infinite collection of sets</ins>? <ins class="diffchange diffchange-inline">Give a proof, or provide a counterexample.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">26</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">17</ins>.''' <ins class="diffchange diffchange-inline">Prove: A subset </ins>$<ins class="diffchange diffchange-inline">F</ins>\<ins class="diffchange diffchange-inline">subseteq </ins>\mathbb{R}$ <ins class="diffchange diffchange-inline">is closed if and only if every Cauchy sequence contained in </ins>$<ins class="diffchange diffchange-inline">F</ins>$ <ins class="diffchange diffchange-inline">converges to an element </ins>in $<ins class="diffchange diffchange-inline">F</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let </del>$<del class="diffchange diffchange-inline">f,g:</del>\<del class="diffchange diffchange-inline">mathbb{R}\to</del>\mathbb{R}$ <del class="diffchange diffchange-inline">be two continuous functions. Define </del>$<del class="diffchange diffchange-inline">h(x)=\max\{f(x),g(x)\}</del>$ <del class="diffchange diffchange-inline">for all $x\</del>in<del class="diffchange diffchange-inline">\mathbb{R}</del>$<del class="diffchange diffchange-inline">. Show that $h$ is continuous on $\mathbb{R}</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">27</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">18</ins>.''' <ins class="diffchange diffchange-inline">Find all accumulation points of the set </ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let $f:</del>\<del class="diffchange diffchange-inline">mathbb</del>{<del class="diffchange diffchange-inline">R</del>}\<del class="diffchange diffchange-inline">to\mathbb</del>{<del class="diffchange diffchange-inline">R</del>}<del class="diffchange diffchange-inline">$ be continuous on $</del>\mathbb{<del class="diffchange diffchange-inline">R</del>}<del class="diffchange diffchange-inline">$, and assume </del>that <del class="diffchange diffchange-inline">for all </del>$\<del class="diffchange diffchange-inline">varepsilon>0$ there is an $N>0$ such that $|f</del>(<del class="diffchange diffchange-inline">x</del>)<del class="diffchange diffchange-inline">|<</del>\<del class="diffchange diffchange-inline">varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $</del>\<del class="diffchange diffchange-inline">mathbb{R}</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\<ins class="diffchange diffchange-inline">[\left\</ins>{<ins class="diffchange diffchange-inline">\frac{1}{m</ins>}<ins class="diffchange diffchange-inline">+</ins>\<ins class="diffchange diffchange-inline">frac</ins>{<ins class="diffchange diffchange-inline">1}{n</ins>}<ins class="diffchange diffchange-inline">\ |\ m,n\in</ins>\mathbb{<ins class="diffchange diffchange-inline">N</ins>}<ins class="diffchange diffchange-inline">\right\}\]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Remember </ins>that $<ins class="diffchange diffchange-inline">A=B</ins>\ <ins class="diffchange diffchange-inline"> \Leftrightarrow\  </ins>(<ins class="diffchange diffchange-inline">A\subseteq B</ins>)\<ins class="diffchange diffchange-inline">wedge (B</ins>\<ins class="diffchange diffchange-inline">subseteq A)</ins>$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">28</del>.'''</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">19</ins>.''' <ins class="diffchange diffchange-inline">Show: If </ins>$<ins class="diffchange diffchange-inline">X</ins>\<ins class="diffchange diffchange-inline">subseteq </ins>\mathbb{R}$ is <ins class="diffchange diffchange-inline">both open and closed, then </ins>$<ins class="diffchange diffchange-inline">X=</ins>\<ins class="diffchange diffchange-inline">mathbb{R}</ins>$ <ins class="diffchange diffchange-inline">or </ins>$<ins class="diffchange diffchange-inline">X=</ins>\<ins class="diffchange diffchange-inline">emptyset</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Let </del>$<del class="diffchange diffchange-inline">f:[a,b]</del>\<del class="diffchange diffchange-inline">to</del>\mathbb{R}$ <del class="diffchange diffchange-inline">be a function. We say $f$ satisfies $(*)$ if there </del>is <del class="diffchange diffchange-inline">an </del>$<del class="diffchange diffchange-inline">M>0$ such that $|f(x)-f(y)|\leq M</del>\<del class="diffchange diffchange-inline">cdot |x-y|</del>$ <del class="diffchange diffchange-inline">for all </del>$<del class="diffchange diffchange-inline">x,y</del>\<del class="diffchange diffchange-inline">in [a,b]</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Let $g</del>:[<del class="diffchange diffchange-inline">0</del>,1<del class="diffchange diffchange-inline">]</del>\<del class="diffchange diffchange-inline">to</del>\<del class="diffchange diffchange-inline">mathbb</del>{<del class="diffchange diffchange-inline">R</del>}<del class="diffchange diffchange-inline">$ be given by $g(x)</del>=\<del class="diffchange diffchange-inline">sqrt</del>{<del class="diffchange diffchange-inline">x</del>}<del class="diffchange diffchange-inline">$. Show that $g$ does not satisfy $(*)$.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''Problem 20.''' Consider the following sets</ins>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Is $g$ uniformly continuous on $</del>[<del class="diffchange diffchange-inline">a</del>,<del class="diffchange diffchange-inline">b</del>]<del class="diffchange diffchange-inline">$? Is $g$ uniformly continuous on $(a</del>,<del class="diffchange diffchange-inline">b)$? Explain!</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">\</ins>[<ins class="diffchange diffchange-inline">A=\left\{1</ins>,<ins class="diffchange diffchange-inline">\frac{</ins>1<ins class="diffchange diffchange-inline">}{2},</ins>\<ins class="diffchange diffchange-inline">frac{1}{3},</ins>\<ins class="diffchange diffchange-inline">frac</ins>{<ins class="diffchange diffchange-inline">1</ins>}<ins class="diffchange diffchange-inline">{4}\ldots\right\},\quad B</ins>=\<ins class="diffchange diffchange-inline">left\</ins>{<ins class="diffchange diffchange-inline">1,\frac{1</ins>}<ins class="diffchange diffchange-inline">{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap</ins>[<ins class="diffchange diffchange-inline">0</ins>,<ins class="diffchange diffchange-inline">1</ins>]<ins class="diffchange diffchange-inline">\]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">For the sets that are compact</ins>, <ins class="diffchange diffchange-inline">explain why. For the other ones, show that they have an open cover without finite subcover.</ins></div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_4&diff=2823&oldid=prevHelmutKnaust at 18:08, 12 November 20192019-11-12T18:08:06Z<p></p>
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<td colspan='2' style="background-color: white; color:black;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 18:08, 12 November 2019</td>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">16</del>.'''  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">25</ins>.'''  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Show</del>: <del class="diffchange diffchange-inline">If </del>$<del class="diffchange diffchange-inline">x$ is </del>an <del class="diffchange diffchange-inline">accumulation point of </del>$<del class="diffchange diffchange-inline">A</del>\<del class="diffchange diffchange-inline">cup B$, then $</del>x<del class="diffchange diffchange-inline">$ is an accumulation point of $A$, or $</del>x$ <del class="diffchange diffchange-inline">is an accumulation point of $B</del>$ (<del class="diffchange diffchange-inline">or both</del>)<del class="diffchange diffchange-inline">. </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let $f</ins>:<ins class="diffchange diffchange-inline">[a,b]\to\mathbb{R}</ins>$ <ins class="diffchange diffchange-inline">be </ins>an <ins class="diffchange diffchange-inline">increasing function. Show that </ins>$\<ins class="diffchange diffchange-inline">lim_{</ins>x<ins class="diffchange diffchange-inline">\to a}f(</ins>x<ins class="diffchange diffchange-inline">)</ins>$ <ins class="diffchange diffchange-inline">exists. What can you say about the relationship between this limit and </ins>$<ins class="diffchange diffchange-inline">f</ins>(<ins class="diffchange diffchange-inline">a</ins>)<ins class="diffchange diffchange-inline">$</ins>?</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"># Does the result also hold for a countably infinite collection of sets</del>? <del class="diffchange diffchange-inline">Give a proof, or provide a counterexample.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">17</del>.''' <del class="diffchange diffchange-inline">Prove: A subset </del>$<del class="diffchange diffchange-inline">F</del>\<del class="diffchange diffchange-inline">subseteq </del>\mathbb{R}$ <del class="diffchange diffchange-inline">is closed if and only if every Cauchy sequence contained in </del>$<del class="diffchange diffchange-inline">F</del>$ <del class="diffchange diffchange-inline">converges to an element </del>in $<del class="diffchange diffchange-inline">F</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">26</ins>.'''  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let </ins>$<ins class="diffchange diffchange-inline">f,g:</ins>\<ins class="diffchange diffchange-inline">mathbb{R}\to</ins>\mathbb{R}$ <ins class="diffchange diffchange-inline">be two continuous functions. Define </ins>$<ins class="diffchange diffchange-inline">h(x)=\max\{f(x),g(x)\}</ins>$ <ins class="diffchange diffchange-inline">for all $x\</ins>in<ins class="diffchange diffchange-inline">\mathbb{R}</ins>$<ins class="diffchange diffchange-inline">. Show that $h$ is continuous on $\mathbb{R}</ins>$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">18</del>.''' <del class="diffchange diffchange-inline">Find all accumulation points of the set </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">27</ins>.'''  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\<del class="diffchange diffchange-inline">[\left\</del>{<del class="diffchange diffchange-inline">\frac{1}{m</del>}<del class="diffchange diffchange-inline">+</del>\<del class="diffchange diffchange-inline">frac</del>{<del class="diffchange diffchange-inline">1}{n</del>}<del class="diffchange diffchange-inline">\ |\ m,n\in</del>\mathbb{<del class="diffchange diffchange-inline">N</del>}<del class="diffchange diffchange-inline">\right\}\]</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let $f:</ins>\<ins class="diffchange diffchange-inline">mathbb</ins>{<ins class="diffchange diffchange-inline">R</ins>}\<ins class="diffchange diffchange-inline">to\mathbb</ins>{<ins class="diffchange diffchange-inline">R</ins>}<ins class="diffchange diffchange-inline">$ be continuous on $</ins>\mathbb{<ins class="diffchange diffchange-inline">R</ins>}<ins class="diffchange diffchange-inline">$, and assume </ins>that <ins class="diffchange diffchange-inline">for all </ins>$\<ins class="diffchange diffchange-inline">varepsilon>0$ there is an $N>0$ such that $|f</ins>(<ins class="diffchange diffchange-inline">x</ins>)<ins class="diffchange diffchange-inline">|<</ins>\<ins class="diffchange diffchange-inline">varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $</ins>\<ins class="diffchange diffchange-inline">mathbb{R}</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Remember </del>that $<del class="diffchange diffchange-inline">A=B</del>\ <del class="diffchange diffchange-inline"> \Leftrightarrow\  </del>(<del class="diffchange diffchange-inline">A\subseteq B</del>)\<del class="diffchange diffchange-inline">wedge (B</del>\<del class="diffchange diffchange-inline">subseteq A)</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem <del class="diffchange diffchange-inline">19</del>.''' <del class="diffchange diffchange-inline">Show: If </del>$<del class="diffchange diffchange-inline">X</del>\<del class="diffchange diffchange-inline">subseteq </del>\mathbb{R}$ is <del class="diffchange diffchange-inline">both open and closed, then </del>$<del class="diffchange diffchange-inline">X=</del>\<del class="diffchange diffchange-inline">mathbb{R}</del>$ <del class="diffchange diffchange-inline">or </del>$<del class="diffchange diffchange-inline">X=</del>\<del class="diffchange diffchange-inline">emptyset</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem <ins class="diffchange diffchange-inline">28</ins>.'''</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Let </ins>$<ins class="diffchange diffchange-inline">f:[a,b]</ins>\<ins class="diffchange diffchange-inline">to</ins>\mathbb{R}$ <ins class="diffchange diffchange-inline">be a function. We say $f$ satisfies $(*)$ if there </ins>is <ins class="diffchange diffchange-inline">an </ins>$<ins class="diffchange diffchange-inline">M>0$ such that $|f(x)-f(y)|</ins>\<ins class="diffchange diffchange-inline">leq M\cdot |x-y|</ins>$ <ins class="diffchange diffchange-inline">for all </ins>$<ins class="diffchange diffchange-inline">x,y</ins>\<ins class="diffchange diffchange-inline">in [a,b]</ins>$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">'''Problem 20.''' Consider the following sets</del>:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Let $g</ins>:[<ins class="diffchange diffchange-inline">0</ins>,1<ins class="diffchange diffchange-inline">]</ins>\<ins class="diffchange diffchange-inline">to</ins>\<ins class="diffchange diffchange-inline">mathbb</ins>{<ins class="diffchange diffchange-inline">R</ins>}<ins class="diffchange diffchange-inline">$ be given by $g(x)</ins>=\<ins class="diffchange diffchange-inline">sqrt</ins>{<ins class="diffchange diffchange-inline">x</ins>}<ins class="diffchange diffchange-inline">$. Show that $g$ does not satisfy $(*)$.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">\</del>[<del class="diffchange diffchange-inline">A=\left\{1</del>,<del class="diffchange diffchange-inline">\frac{</del>1<del class="diffchange diffchange-inline">}{2},</del>\<del class="diffchange diffchange-inline">frac{1}{3},</del>\<del class="diffchange diffchange-inline">frac</del>{<del class="diffchange diffchange-inline">1</del>}<del class="diffchange diffchange-inline">{4}\ldots\right\},\quad B</del>=\<del class="diffchange diffchange-inline">left\</del>{<del class="diffchange diffchange-inline">1,\frac{1</del>}<del class="diffchange diffchange-inline">{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap</del>[<del class="diffchange diffchange-inline">0</del>,<del class="diffchange diffchange-inline">1</del>]<del class="diffchange diffchange-inline">\]</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"># Is $g$ uniformly continuous on $</ins>[<ins class="diffchange diffchange-inline">a</ins>,<ins class="diffchange diffchange-inline">b</ins>]<ins class="diffchange diffchange-inline">$? Is $g$ uniformly continuous on $(a</ins>,<ins class="diffchange diffchange-inline">b)$? Explain!</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">For the sets that are compact</del>, <del class="diffchange diffchange-inline">explain why. For the other ones, show that they have an open cover without finite subcover.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_4&diff=2776&oldid=prevHelmutKnaust: Created page with "'''Problem 16.''' # Show: If $x$ is an accumulation point of $A\cup B$, then $x$ is an accumulation point of $A$, or $x$ is an accumulation point of $B$ (or both). # Does th..."2019-10-09T18:42:07Z<p>Created page with "'''Problem 16.''' # Show: If $x$ is an accumulation point of $A\cup B$, then $x$ is an accumulation point of $A$, or $x$ is an accumulation point of $B$ (or both). # Does th..."</p>
<p><b>New page</b></p><div>'''Problem 16.''' <br />
# Show: If $x$ is an accumulation point of $A\cup B$, then $x$ is an accumulation point of $A$, or $x$ is an accumulation point of $B$ (or both). <br />
# Does the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.<br />
<br />
'''Problem 17.''' Prove: A subset $F\subseteq \mathbb{R}$ is closed if and only if every Cauchy sequence contained in $F$ converges to an element in $F$.<br />
<br />
'''Problem 18.''' Find all accumulation points of the set <br />
\[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\]<br />
Remember that $A=B\ \Leftrightarrow\ (A\subseteq B)\wedge (B\subseteq A)$.<br />
<br />
'''Problem 19.''' Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.<br />
<br />
'''Problem 20.''' Consider the following sets:<br />
\[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\]<br />
For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.</div>HelmutKnaust