I need help with the following problem:
Let $(X,\rho)$ be a compact metric space. Prove that if $K$ is a compact subset of $C(X)=C(X,\mathbb{R})$ (i.e. continuous functions with real values) whose linear span is dense in $C(X)$, then the pseudometric $d(x,y)= \sup _{f \in K} \lvert f(x) - f(y)\rvert$ on $X$ is actually a metric. Moreover, show that $d$ and $\rho$ generate the same topology.
The existence of $K$ was proved in another thread here: Dense subset of $C(X)$.
Thanks.