Uniform Cauchy sequence

A Cauchy condition for function sequences with a uniform bound over the domain.

Uniform Cauchy sequence

A sequence of functions $(f_n):X\to Y$ into a metric space $(Y,d)$ is uniform Cauchy on $X$ if for every $\varepsilon>0$ there exists $N$ such that for all $m,n\ge N$ and all $x\in X$ ,

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

Equivalently,

\sup_{x\in X} d\bigl(f_m(x),f_n(x)\bigr)<\varepsilon \quad \text{for all } m,n\ge N.

This is the Cauchy formulation of uniform convergence ; their relationship is summarized by uniform Cauchy if and only if uniform convergence (under the standard completeness hypotheses on the codomain). In the real-valued bounded setting, this is exactly the Cauchy condition in the uniform metric .

Examples:

On $[0,1]$ , $f_n(x)=x/n$ is uniform Cauchy because $\sup_{x\in[0,1]}|f_m(x)-f_n(x)|\le |1/m-1/n|$ .
On $[0,1]$ , $f_n(x)=x^n$ is not uniform Cauchy (the functions separate near $x=1$ for large indices).