compact sets 2,

Is my argument rigorous enough:

Prove the intersection of an arbitrary collection of compact subset of metric space M is compact.

Let B be the intersection of all the compact subsets.

Let (p_n) be a sequence in B. Then (p_n) is a sequence in each subset A in the intersection. But since every A compact, (p_n) has an accumulation point, say p, in every A. So p in B. Now the book doesn't prove it, but it gives a reference that compact set S in a metric space is equivalent to every sequence in S having an accumulation point in S. So by this fact, we can conclude B is compact.
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