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\centerline {\bf Math 525a - Fall 17 - Homework 3} 

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\no 1. Problems 4(c) and 7(c) in section 2.1 of the text. 

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\no 2. Problem 9 in section 2.1 of the text. 

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\no 3. Let $f$ be a real-valued function defined on $\reals$. Suppose 
that for all $x_0$, $\lim_{x \rightarrow x_0}$ exists. Prove that for every 
interval $[a,b]$, $f$ is bounded on the interval, i.e., there an $M$ 
such that $|f(x)| \le M$ for $x \in [a,b]$.  Hint: use the previous problem 
to construct an open cover of $[a,b]$.

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\no 4. Problem 12 in section 2.1 of the text. 

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\no 5. Let $K$ be a compact subset of the real numbers, and let $F$ be 
a closed subset of the real numbers. Suppose that $K$ and $F$ are disjoint. 
Prove there is an $\epsilon>0$ such that for all $x \in K$ and all $y \in F$, 
$|x-y| \ge \epsilon$. Hint: Since $F$ is closed, if $x \notin F$ there is an 
$\epsilon>0$ such that $(x-\epsilon,x+\epsilon)$ is disjoint from $F$. Use 
this to construct an open cover of $K$. 

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