CRN 11247: HW 4
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.
