CRN 11378: HW 4

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'''Problem 25.'''  
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'''Problem 16.'''  
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Let $f:[a,b]\to\mathbb{R}$ be an increasing function. Show that $\lim_{x\to a}f(x)$ exists. What can you say about the relationship between this limit and $f(a)$?
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# 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).
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# Does the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.
  
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'''Problem 26.'''  
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'''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 $F$.
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Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that $h$ is continuous on $\mathbb{R}$.
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'''Problem 27.'''  
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'''Problem 18.''' Find all accumulation points of the set
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Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$, and assume that for all $\varepsilon>0$ there is an $N>0$ such that $|f(x)|<\varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $\mathbb{R}$.
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\[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\]
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Remember that $A=B\ \Leftrightarrow\  (A\subseteq B)\wedge (B\subseteq A)$.
  
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'''Problem 28.'''
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'''Problem 19.''' Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.
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Let $f:[a,b]\to\mathbb{R}$ be a function. We say $f$ satisfies $(*)$ if there is an $M>0$ such that $|f(x)-f(y)|\leq M\cdot |x-y|$ for all $x,y\in [a,b]$.
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# Let $g:[0,1]\to\mathbb{R}$ be given by $g(x)=\sqrt{x}$. Show that $g$ does not satisfy $(*)$.
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'''Problem 20.''' Consider the following sets:
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# Is $g$ uniformly continuous on $[a,b]$? Is $g$ uniformly continuous on $(a,b)$? Explain!
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\[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\]
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For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.

Revision as of 12:11, 12 November 2019

Problem 16.

  1. 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).
  2. Does the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.

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 $F$.

Problem 18. Find all accumulation points of the set \[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\] Remember that $A=B\ \Leftrightarrow\ (A\subseteq B)\wedge (B\subseteq A)$.

Problem 19. Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.

Problem 20. Consider the following sets: \[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\] For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.

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