Compact metric space criteria

Let $(X, \rho)$ - metric-space. Then the following propositions are equivalent:

1. X - a compact-space
2. X - complete-metric-space and totally-bounded-space
3. $\forall {x_n} \subset X,,\exists {x_k} \subset {x_n} \hookrightarrow x_k \rightarrow x^*$
4. $\forall$ non-finite set $\exists$ sequential-point-of-closure
5. Continuous functions $X \rightarrow \mathbb{R}$ are the subset of bounded functions $X \rightarrow \mathbb{R}$