I've had issues solving the following problem. The closest I have come is shown below. The problem is that in my "proof", I'm not using the fact that $K$ is compact, which makes me suspicious of the validity of my proof. Any help is highly appreciated.
Problem
Let $X$ be a compact space. Consider a compact subset $K \subset \mathbb{R}$ and let $(f_n)_{n\ge 1}$ be a seq. of continuous functions where $f_n : X \to K$ such that $f_n \to f$ pointwise. Show that if the sequence $(f_n)$ is equicontinuous, then $f$ is also continuous and the convergence is uniform.
Proof
Since $f_n$ converges pointwise, the sequence is pointwise bounded. By Arzela's theorem, there exists a subsequence $(f_{n_k})$ that converges uniformly to $f^*$. By the uniform limit theorem, $f^*$ is continuous. Given $x\in X$, $f_n(x) \to f(x)$, hence any convergence subsequence of $(f_n(x))$ must converge to $f(x)$. Hence $f^* = f$.