$C(K,\mathbb{R})$ is complete in the supremum metric for every nonempty compact metric space $K$

Given: A nonempty compact metric space $(K,d)$ and the set $C(K,\mathbb{R})$ of continuous real-valued functions on it.

Every continuous real-valued function on a nonempty compact metric space has a bounded range (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

If $S$ is nonempty, the formula $d_{\mathrm{\infty }}(f,g)={\sup }_{x\in S}\mathrm{\mid }f(x)-g(x)\mathrm{\mid }$ defines a metric on the set of bounded functions $S\to \mathbb{R}$ (The supremum metric $d_{\mathrm{\infty }}(f,g)={\sup }_x\mathrm{\mid }f(x)-g(x)\mathrm{\mid }$ is a metric on the bounded real-valued functions on a nonempty set).

A sequence is Cauchy in a metric $d$ when, for every positive error, all pairwise distances sufficiently far out are below that error; it converges to $p$ when its distances to $p$ tend to zero (Cauchy sequence in a metric space, Convergence of a sequence in a metric space: $x_k\to x$ iff $d(x_k,x)\to 0$ in $\mathbb{R}$).

A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy (A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy).

A uniform limit of continuous real-valued functions on a metric space is continuous (The uniform limit of continuous real-valued functions on a metric space is continuous).

A metric space is complete when every Cauchy sequence in it converges to one of its points (Complete metric space: every Cauchy sequence converges in the space).