Let $X$ be a compact topological space and let $K\subset \mathbb{R}^n$ be compact. Let $F = \{f_n\}_{n \ge 1}$ be a sequence of continuous functions where $f_n : X \to K$ that converges pointwise to $f$.
Show that if $F$ is equicontinuous, then $f$ is continuous and $f_n$ converges uniformly to $f$.
Attempt
a) It is enough to show that the convergence is uniform, since the Uniform Limit Theorem then implies that $f$ is continuous, correct?
b) Since the convergence is pointwise, $F$ must be pointwise bounded, correct? (If it's not, then I have a big misunderstanding...)
c) By Arzela's theorem, since $X$ is compact,$F$ pointwise bounded and equicontinuous, $F$ has a uniformly convergent subsequence, correct?
d) Since $F$ has a uniformly convergent subsequence, can I conclude that it converges uniformly to $f$? Why? Why not?
Any help is highly appreciated.