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MATH 4630‑01 Homework Assignment #4
DUE: Friday, Mar 27, 11:59p Sections 3.3, 3.4
Instructions: Show all work clearly and justify each conclusion. Collaboration is encouraged, but write up your solutions individually in your own words.
For any prove/disprove problem: either give a proof, or give a specific counterexample (with a brief explanation of why it works).
  1. For each set $K$, determine whether it is compact. If not compact, produce a sequence $(x_n) \subseteq K$ such that every convergent subsequence converges to a limit not in $K$. Do not use the Heine–Borel Theorem.
    1. $\mathbb{N}$.
    2. $\left\{1,\, \frac{1}{2},\, \frac{2}{3},\, \frac{3}{4},\, \frac{4}{5},\, \ldots\right\}$.
    3. $\left\{1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2} : n \in \mathbb{N}\right\}$. (Hint: you may use $\sum_{n=1}^\infty 1/n^2 = \pi^2/6$.)
  2. Assume $K$ is compact and $F$ is closed. For each set, decide whether it is definitely compact, definitely closed, both, or neither.
    1. $K \cap F$.
    2. $\overline{F^c \cup K^c}$.
    3. $K \setminus F = \{x \in K : x \notin F\}$.
    4. $\overline{K \cap F^c}$.
  3. Let $A$ and $B$ be nonempty subsets of $\mathbb{R}$. Show that if there exist disjoint open sets $U$ and $V$ with $A \subseteq U$ and $B \subseteq V$, then $A$ and $B$ are separated.

    Hint: First show that $\overline{A} \subseteq U$ and $\overline{B} \subseteq V$.

  4. Let $K_1 \supseteq K_2 \supseteq K_3 \supseteq \cdots$ be a nested sequence of nonempty compact sets. Show that $\bigcap_{n=1}^{\infty} K_n$ is nonempty.

    Hint: For each $n$, choose $x_n \in K_n$ and consider the sequence $(x_n)$.

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