Uniform Cauchy implies uniform convergence

Uniform Cauchy implies uniform convergence: Let $X$ be a set and let $(Y,d)$ be a complete metric space . If $(f_n)$ is a uniformly Cauchy sequence of functions $f_n:X\to Y$, then there exists a function $f:X\to Y$ such that $f_n\to f$ uniformly on $X$.

This lemma is the completeness principle for uniform convergence: the space of bounded functions with the sup metric is complete when the target is complete.