HelmutKnaust: 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..."

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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..."

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'''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 the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.

'''Problem 17.''' A Cauchy sequence $(a_n)$ is said to be ''positive'', if for all $k\in\mathbb{N}$ there is an $N\in\mathbb{N}$ such that $a_n>-\frac{1}{k}$ for all $n\geq N$.
#Show that the sum of two positive Cauchy sequences is positive.
#Show that the product of two positive Cauchy sequences is positive.

'''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.

CRN 11247: HW 4 - Revision history

HelmutKnaust: 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..."