Homework 6

Metrics and Metric Spaces

1. Complete problem 24 in the text (Chapter 2).

Convergence and Limits

2. Prove or give a counterexample. Let $a_n$ be a sequence such that $|a_{n+1}-a_n|$ converges to zero. Does $a_n$ have to converge?

3. Use Theorem 2 on page 55 to show that the function

(1)
\begin{align} f(x)=\Big\{\begin{matrix}x\ & x\geq 0 \\ 1 & x<0\end{matrix} \end{align}

is not continuous. In particular find a sequence $a_n$ which converges to 0 but for which $f(a_n)$ does not converge to $f(0)$.

4. Prove that if real sequences $a_n$ and $b_n$ converge to $a$ and $b$, then the sequence of products $a_n\cdot b_n$ converges to $a\cdot b$.

5. Use Theorem 2 to show that the product of two continuous functions is continuous. That is, prove that if f and g are continuous, then $h(x)=f(x)\cdot g(x)$ is continuous. (Note that you will also need problem 4 above).

Open and Closed Sets

6. Say whether the following subsets of $\mathbb{R}$ are open, closed, neither, or both. Give reasons. (a) $[0,1)$; (b) $\mathbb{Z}$; (c) $\{x\in\mathbb{R}:\sin(x)>0\}$; (d) the union of sets $[1/n,1)$ for all integers $n\geq2$, otherwise written $\bigcup_{n=2}^\infty [1/n,1)$.

7. Show by example that the union of infinitely many closed sets need not be closed.

8. Complete problem 3 in the text (Chapter 2).