In mathematics, a sequence of functions ${\displaystyle \{f_{n}\}}$ from a set S to a metric space M is said to be uniformly Cauchy if:

• For all ${\displaystyle \varepsilon >0}$, there exists ${\displaystyle N>0}$ such that for all ${\displaystyle x\in S}$: ${\displaystyle d(f_{n}(x),f_{m}(x))<\varepsilon }$ whenever ${\displaystyle m,n>N}$.

Another way of saying this is that ${\displaystyle d_{u}(f_{n},f_{m})\to 0}$ as ${\displaystyle m,n\to \infty }$, where the uniform distance ${\displaystyle d_{u}}$ between two functions is defined by

${\displaystyle d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).}$