Sequences of Real-Valued Maps: the Map of Pointwise Limits on a Subset, and the Sequence of Pointwise Partial Sums

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $D$ be a set and $S$ a subset of $D$, let $\mathbb{R}^{D}$ be the set of maps from $D$ to $\mathbb{R}$ of Sets and Maps: Ordinary Notation (v), and let sequences be as in Sequences (i). For a sequence $(f_{k})_{k\in\mathbb{N}}$ in $\mathbb{R}^{D}$ and $p\in D$, $(f_{k}(p))_{k}$ is the sequence $k\mapsto f_{k}(p)$ in $\mathbb{R}$ of The Sequences k ↦ f_k(p) of a Sequence in ℝ^I, and the Map of Their Limits (i).

(i) Let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{D}$ such that $(f_{k}(p))_{k}$ is convergent for every $p\in S$. Then there is exactly one map $L:S\to\mathbb{R}$ such that $L(p)$ is the limit $\lim_{k\to\infty}f_{k}(p)$ for every $p\in S$.

(ii) Let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{D}$. There is exactly one sequence $(s_{n})_{n\in\mathbb{N}}$ in $\mathbb{R}^{D}$ such that $s_{n}(x)$ is the finite sum $\sum_{k=1}^{n}g_{k}(x)$ for all $n\in\mathbb{N}$ and $x\in D$. For every $x\in D$, the sequence $(s_{n}(x))_{n}$ is the sequence of partial sums of $(g_{k}(x))_{k}$.