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Mathematical Analysis Fundamentals by Agamirza Bashirov

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Theorem 4.58

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

(a) Eimage is compact.

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

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

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

Proof

For the implication (a)(b), assume ...

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