Every compact metric space is complete I need to prove that every compact metric space is complete. I think I need to use the following two facts: 


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*A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with finite intersection property has $\bigcap\{F:F\in\mathcal{F}\}\neq\emptyset$.

*A metric space $(X,d)$ is complete if and only if for any sequence $\{F_n\}$ of non-empty closed sets with $F_1\supset F_2\supset\cdots$ and $\text{diam}~F_n\rightarrow0$, $\bigcap_{n=1}^{\infty}F_n$ contains a single point.


I do not know how to arrive at my result that every compact metric space is complete. Any help? 
Thanks in advance.

Here is a simple proof:
Note that a compact metric space is sequentially compact.
So that every sequence, including Cauchy sequences, have convergent subsequences.
Since all Cauchy sequences have convergent subsequences, we must find that all Cauchy sequences converge. Meaning that the metric space is complete.
