= Uniform Cauchy criterion
{c}
{title2=$\sup_x|f_n(x)-f_m(x)|\to0\quad(m,n\to\infty)$}
A <sequence of functions> $f_n:E\to\mathbb R$ has a <uniform limit> if and only if, for every $\varepsilon>0$, there is $N$ such that $|f_n(x)-f_m(x)|<\varepsilon$ for every $m,n\geq N$ and every $x\in E$. Necessity follows from the <triangle inequality>. For sufficiency, <completeness of the real numbers> first gives a <pointwise limit> $f$. Letting $m\to\infty$ in the bound with tolerance $\varepsilon/2$ gives $|f_n(x)-f(x)|\leq\varepsilon/2$ uniformly in $x$. The same proof works for a <complete metric space> as codomain; completeness is essential.
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