---
tag: 01XAY17QA3
kind: definition
name: Uniformly Cauchy sequence of functions
msc: [40A30]
defines: 0BMY28TBSF
uses: [0QZMTN5AXX, 08HJRTG5N7, 0ZA4AX4MW3]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_convergence
    locator: Properties
---

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$.
The sequence $(f_n)$ is uniformly Cauchy on $S$ if for every $\varepsilon > 0$ there is $N$ such that
$$|f_n(x) - f_m(x)| < \varepsilon \quad \text{for all } m, n \ge N \text{ and all } x \in S.$$

## Remarks

For each $x$, the sequence $(f_n(x))$ is then a [[Cauchy sequence|0QZMTN5AXX]]. So it has a limit $f(x)$, by the [[Cauchy criterion|0ZA4AX4MW3]].
A sequence of real functions is uniformly Cauchy on $S$ if and only if it [[converges uniformly|08HJRTG5N7]] on $S$.
This criterion proves uniform convergence without knowing the limit. Mathlib defines it as `UniformCauchySeqOn`.

## Explanations

### L1 human

A uniformly Cauchy sequence has graphs that get close to each other, uniformly in $x$.
Such a sequence always converges uniformly to some function.