I am trying to solve the following problem in preparation for an upcoming exam in introductory topology and I’m unable to complete it entirely. My problem is with (b), but I’ve stated the full problem since it is likely that (a) is required for (b):
Let $X$ be a compact metric space.
a. Prove that every continuous function $f:Xrightarrow mathbb{R}$ is uniformly continuous.
b. Let $T,S : X rightarrow X$ be isometries.
Let $f_1 in C(X,mathbb{R})$. Denote $f_{n+1} = frac{1}{2}Big(f_n(T(x))+f_n(S(x))Big) quadforall xin X, nin mathbb{N}$ .
Prove that ${f_n}_{nin mathbb{N}}$ has a converging subsequence using the sup norm.
So I’ve tried the following two approaches:

Use ArzelaAscoli by first proving $F:= {f_n}$ is closed, bounded and equicontinuous. It seems impossible to prove it is closed – maybe I’m missing something? (tried assuming by contradiction there is a $g$ in $cl(F)$ not in $F$ but this didn’t take me too far).

Proving the space is complete (since $X$ is compact $C(X,mathbb{R})= B_c(X,mathbb{R})$ and using the fact that $X$ is a topological space and $mathbb{R}$ is complete) and then trying to prove that ${f_n}_{nin mathbb{N}}$ is Cauchy, therefore converges, therefore has a convergent subsequence. Also got stuck in this case, although it seems more promising due to the isometries, but not sure.
What am I missing? Any help would be much appreciated!