CRN 11247: HW 4

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. 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 $n\geq N$.

1. Show that the sum of two positive Cauchy sequences is positive.
2. 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.