Q:

Dense subset of $C(X)$

I am trying to prove the next proposition, but i need some help:
Let $ (X,\rho)$ be compact metric space. Prove that there exist a compact subset $K$ of $C(X)$ whose linear span is dense in $C(X)$
Thanks.

A:

Hints:

$C(X)$ is separable.
Given any sequence in a Banach space, you can rescale the terms of the sequence so that it converges to 0.
In any metric space. if $x_n \to x$, then $\{x_n\} \cup \{x\}$ is compact. 

