If $X$ is a compact topological space and if some sequence $\{f_n\}$ of continuous functions separates points on $X$, then $X$ is metrizable

I have a little problem with this Theorem 3.8 c) If $X$ is a compact topological space and if some sequence $\{f_n\}$ of continuous functions separates points on $X$, then $X$ is metrizable.

The big deal is that in the proof Rudin constructs a metric $d(x, y)=\sum_{n=1}^{\infty}2^{-n}|f_n(x)-f_n(y)|$, and this metric induces a topology denoted by $\tau_{d}$ and I have to prove that $\tau_{d}\subseteq \tau$ so in order to do that I must take a basic $B_r(x) \in \tau_{d}$ and prove that there exists an open set $U \in \tau$ such that $U\subseteq B_r(x)$ but I don't know exactly how to do it, may you give any hint please? I'm really stuck.

I just have some ideas with weak topology but I'm not sure if it works.

Because $d$ is continuous as a function on $X \times X$, $\tau_d \subseteq \tau$ is immediate:

$B_r(x)=(d_x)^{-1}[(-\infty,r)]$ where (for any fixed $x$) the map $d_x: y \to d(x,y)$ is continuous from $X$ to $\Bbb R$.