(a) State the Arzela-Ascoli theorem, explaining the meaning of all concepts involved.

(b) Prove the Arzela-Ascoli theorem.

(c) Let $K$ be a compact topological space. Let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a sequence in the Banach space $C(K)$ of real-valued continuous functions over $K$ equipped with the supremum norm $\|\cdot\|$. Assume that for every $x \in K$, the sequence $f_{n}(x)$ is monotone increasing and that $f_{n}(x) \rightarrow f(x)$ for some $f \in C(K)$. Show that $\left\|f_{n}-f\right\| \rightarrow 0$ as $n \rightarrow \infty$.