Recall the Heine-Borel Theorem: Every closed and bounded subset of R^m is compact. However, a closed and bounded subset

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Recall the Heine-Borel Theorem: Every closed and bounded subset
of R^m is compact. However, a closed and bounded subset of a
general metric space may fail to be compact. For example, the set N
of natural numbers equipped with the discrete metric (i) is a
complete metric space; (ii) it is closed as a subset of itself;
(iii) and it is bounded; (iv) but it is not compact.Prove the
statements (i), (ii), (iii), and (iv).
Given a set M, define the distance between distinct points of M
to be 1 and the distance between every point and itself to be 0.
This is the discrete metric.