# Statements Let $X$ be a non-empty [[Topology|topological space]] and let $C_{b}(X)$ denote the set of [[Bounded Real Function|bounded]], [[Continuity|continuous]] functions $X\to \mathbb{R}$. Then $(C_{b}(X),\lVert \cdot \rVert_{\infty})$ is a [[Complete metric spaces|complete metric space]]. # Proofs ###### Proof when $X$ is a metric space Let $f_{n}$ be a Cauchy sequence in $C_{b}(X)$. Since, $B(X)$, the set of bounded functions $f:X\to \mathbb{R}$ which [[Completeness of the supremum norm on bounded real-valued functions on non-empty set|is complete wrt to sup norm]], we have that $f_{n}\to f \in B(X)$. Furthermore, by [[Uniform limit of continuous real-valued functions is continuous]], we have that $f$ is continuous which completes the proof. ###### Sketch of proof when $X$ is topological space