Pointwise and Uniform Convergence of a Sequence 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 $S\subseteq D$, 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), and let $f:E\to\mathbb{R}$ be a map on a set $E$ with $S\subseteq E$. 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), with $k$-th term $f_{k}(x)$.

(i) $(f_{k})_{k\in\mathbb{N}}$ converges pointwise to $f$ on $S$ if for every $x\in S$ the sequence $(f_{k}(x))_{k\in\mathbb{N}}$ converges to $f(x)$: for every real $\varepsilon>0$ there is $K\in\mathbb{N}$ such that $|f_{k}(x)-f(x)|<\varepsilon$ for every $k\in\mathbb{N}$ with $k\ge K$.

(ii) $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $S$ if for every real $\varepsilon>0$ there is $K\in\mathbb{N}$ such that $|f_{k}(x)-f(x)|<\varepsilon$ for every $k\in\mathbb{N}$ with $k\ge K$ and every $x\in S$.