Uniform Cauchy criterion

For functions into a complete metric space, uniform convergence is equivalent to being uniformly Cauchy.

Uniform Cauchy criterion

Uniform Cauchy criterion: Let $E$ be a set and let $(Y,\rho)$ be a complete metric space . A sequence of functions $f_n:E\to Y$ converges uniformly on $E$ if and only if it is uniformly Cauchy , meaning: for every $\varepsilon>0$ there exists $N$ such that for all $m,n\ge N$ and all $x\in E$ ,

\rho\bigl(f_n(x),f_m(x)\bigr)<\varepsilon.

Equivalently, uniform convergence is exactly convergence in the uniform metric (and, for bounded real-valued functions, in the supremum norm ).