Question 6.T.13: For any set E ⊆ R, the following statements are equivalent: ...
For any set E ⊆ \mathbb{R}, the following statements are equivalent:
(i) E is compact.
(ii) E is closed and bounded.
(iii) Every sequence in E has a subsequence which converges to a point in E.
The equivalence of (i) and (ii) is the content of Heine-Borel’s theorem, so we need only prove the equivalence of (ii) and (iii). To prove that (ii) implies (iii), let (x_{n}) be a sequence in the closed and bounded set E. Being bounded, the sequence (x_{n}) has a convergent subsequence (x_{n_{k}}) by Theorem 3.14. Since E is closed, \lim x_{n_{k}} lies in E by Theorem 3.20.
In the other direction, suppose statement (iii) is true. If E were unbounded, we would be able to form a sequence (x_{n}) in E such that
|x_{n}| > n for all n ∈ \mathbb{N}.
But such a sequence would have no convergent subsequence, thereby violating (iii). Hence E is bounded. On the other hand, let (x_{n}) be a sequence in E such that x_{n} → x. (iii) implies (x_{n}) has a subsequence which converges to a point in E. But such a subsequence can only converge to x. Hence x ∈ E and, by Theorem 3.20, E is closed.