The Uniform Cauchy Criterion for Sequences of Real-Valued Maps

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $D$ be a set, let $S\subseteq D$ and let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in the set $\mathbb{R}^{D}$ of maps from $D$ to $\mathbb{R}$ of Sets and Maps: Ordinary Notation (v). Pointwise and uniform convergence on $S$ to a map $g:S\to\mathbb{R}$ are as in Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (i) and Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii), read with $E=S$ and $g$ in place of $f$, and for $x\in S$, $(f_{k}(x))_{k\in\mathbb{N}}$ is the sequence in $\mathbb{R}$ of The Sequences k ↦ f_k(p) of a Sequence in ℝ^I, and the Map of Their Limits (i).

(i) $(f_{k})_{k\in\mathbb{N}}$ converges uniformly on $S$ to some map $f:S\to\mathbb{R}$ if and only if for every real $\varepsilon>0$ there is $K\in\mathbb{N}$ such that $|f_{m}(x)-f_{k}(x)|<\varepsilon$ for all $m,k\in\mathbb{N}$ with $m\ge K$ and $k\ge K$ and every $x\in S$.

(ii) If $(f_{k})_{k\in\mathbb{N}}$ converges uniformly on $S$ to some map from $S$ to $\mathbb{R}$, then for every $x\in S$ the sequence $(f_{k}(x))_{k\in\mathbb{N}}$ is convergent; $(f_{k})_{k\in\mathbb{N}}$ converges uniformly on $S$ to the map $x\mapsto\lim_{k\to\infty}f_{k}(x)$ from $S$ to $\mathbb{R}$, with the limit taken at each $x$; and this map is the only map $g:S\to\mathbb{R}$ such that $(f_{k})_{k\in\mathbb{N}}$ converges pointwise to $g$ on $S$, and the only one such that it converges uniformly to $g$ on $S$.