Uniformly Cauchy

$u_n(x)$ in $x\in A$ is said to be uniformly Cauchy iff:

\forall \epsilon \gt 0 \exists N \in \mathbb{Z}^+ \forall m,n \gt N \forall x \in A; \lvert u_n(x)-u_m(x) \rvert \lt \epsilon

If $u_n(x)$ is a sequence of real-valued functions, then:

u_n(x)\text{ converges uniformly} \iff u_n(x)\text{ is uniformly Cauchy}

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October 17, 2026 1 min read

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