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Theorem 4.58

If a metric space $E$ has one of the following properties, then it has the other three:

(a) $E$ is compact.

(b) Every nested sequence of nonempty closed sets in $E$ has a nonempty intersection.

(c) Every infinite subset of $E$ has a limit point.

(d) Every sequence in $E$ has a convergent subsequence.

Proof

For the implication $\left(\text{a}\right)⇒\left(\text{b}\right)$, assume ...

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