Equivalent statements to compactness of a metric space

Theorem statement

Given a metric space [ilmath](X,d)[/ilmath], the following are equivalent:

1. [ilmath]X[/ilmath] is compact
2. Every sequence in [ilmath]X[/ilmath] has a subsequence that converges (AKA: having a convergent subsequence)
3. [ilmath]X[/ilmath] is totally bounded and complete

Proof

[ilmath]1)\implies 2)[/ilmath]: [ilmath]X[/ilmath] is compact [ilmath]\implies[/ilmath] [ilmath]\forall(a_n)_{n=1}^\infty\subseteq X\ \exists[/ilmath] a sub-sequence [ilmath](a_{k_n})_{n=1}^\infty[/ilmath] that coverges in [ilmath]X[/ilmath]

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Equivalent statements to compactness of a metric space/1-implies-2 proof

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TODO: Rest

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