Uniformly Cauchy sequence

In mathematics, a sequence of functions is said to be uniformly Cauchy if the sequence of their uniform distances is a Cauchy sequence of real numbers.

In more detail, let $(X,{\mathcal {T}})$ be a topological space and let $(Y,d)$ be a metric space. Recall that the uniform distance between two functions $f,g:X\to Y$ is defined by

$d_{\infty }(f,g):=\sup _{x\in X}d(f(x),g(x)).$

If $f_{n}:X\to Y$ for each natural number $n$, then the sequence $(f_{n})$ is said to be uniformly Cauchy if, for all $\varepsilon >0$, there is some natural number $M$ such that $d_{\infty }(f_{n},f_{m})<\varepsilon$ whenever $m,n\geq M$, i.e.

$d_{\infty }(f_{n},f_{m})\to 0$ as $m,n\to \infty .$