Sections of a Sequence of Maps into a Metric Space, and the Map of Their Limits

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(Y,d_{Y})$ be a metric space, let $D$ be a set, let $Y^{D}$ be the set of maps from $D$ to $Y$ of Sets and Maps: Ordinary Notation (v), and let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in $Y^{D}$.

(i) For every $t\in D$ there is exactly one sequence in $Y$ whose $k$-th term is $f_{k}(t)$ for every $k\in\mathbb{N}$; it is written $(f_{k}(t))_{k\in\mathbb{N}}$.

(ii) Let $S\subseteq D$ be such that, for every $t\in S$, the sequence $(f_{k}(t))_{k\in\mathbb{N}}$ is convergent in $(Y,d_{Y})$. Then there is exactly one map $f:S\to Y$ such that, for every $t\in S$, $(f_{k}(t))_{k\in\mathbb{N}}$ converges to $f(t)$ in $(Y,d_{Y})$; and, for every $t\in S$, $f(t)$ is the limit $\lim_{k\to\infty}f_{k}(t)$ of $(f_{k}(t))_{k\in\mathbb{N}}$ in $(Y,d_{Y})$.