Completeness of C(K) under the sup norm

Let $(K,d)$ be a compact metric space and let $C(K,\mathbb{R})$ denote the set of continuous functions $f:K\to\mathbb{R}$.

Define the sup norm by $\|f\|_\infty=\sup_{x\in K}|f(x)|,$ and the induced metric $d_\infty(f,g)=\|f-g\|_\infty=\sup_{x\in K}|f(x)-g(x)|.$

Theorem: The metric space $\bigl(C(K,\mathbb{R}),d_\infty\bigr)$ is complete .

This is the basic Banach-space fact behind many compactness and approximation arguments: uniform Cauchy sequences of continuous functions converge uniformly to a continuous function.