TITLE: Every compact set $A \subset \mathbb{R}$ is bounded
QUESTION [1 upvotes]: Does this proof work?
Pf(contrapositive)
Suppose A is not bounded.
Then $\mathbb{R} - A$ is bounded
Thus for some $p \in \mathbb{R}-A$, and for every $\epsilon > 0$, the interval
$(p-\epsilon, p+\epsilon) \cap A \neq \emptyset$
Let $U_n = \mathbb{R} - [p - \frac{1}{n}, p + \frac{1}{n}]$
$A \subset \cup^{n=1}_{\infty}U_n$ is bounded

REPLY [1 votes]: You need to use a cover of the real numbers that consists of bounded sets only, like the one suggested by Mirko. Compactness implies that a finite number of these bounded sets cover $A.$
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