Equivalent statements to compactness of a metric space

Theorem statement

Given a metric space $(X,d)$, the following are equivalent:

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

Proof

[Expand]

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

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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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