In the mafred stoll, Real analysis, p.326 ( 8.2.6 Theorem ) he state that
8.2.5 Theorem. Suppose the sequence $\{f_n\}$ of real-valued functions on the set $E$ converges pointwise to $f$ on $E$. For each $n\in \mathbb{N}$, set
$$ M_n:= \sup_{x\in E}|f_n(X) -f(x)|$$
Then $\{f_n\}$ converges unifomly to $f$ on $E$ if and only if $\lim_{n\to \infty} M_n =0$.
Q. Perhaps, the theorem also holds when we replace $E$ by arbitrary (sub) metric space ? Definition of uniform convergence on metric space is as https://proofwiki.org/wiki/Definition:Uniform_Convergence/Metric_Space. I can't find any related information.