Let $X$ be a nonempty [topological space](/page/Topological%20Space), and let $C_b(X)$ be the real [vector space](/page/Vector%20Space) of all bounded continuous functions $f:X \to \mathbb{R}$. Equip $C_b(X)$ with the sup norm $\|\cdot\|_\infty:C_b(X)\to [0,\infty)$ defined by
Then every [Cauchy sequence](/page/Cauchy%20Sequence) in the normed space $(C_b(X),\|\cdot\|_\infty)$ converges in $\|\cdot\|_\infty$ to an element of $C_b(X)$. Equivalently, $(C_b(X),\|\cdot\|_\infty)$ is complete.