Sums of Series: a Convergent Series Has a Real Sum, and a Bound on All Partial Sums Bounds the Sum in [0,∞]

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $[-\infty,\infty]$ and $[0,\infty]$ be as in The Extended Real Line, Its Order, and [0,∞] (ii), with the order $\le$ of The Extended Real Line, Its Order, and [0,∞] (iii).

(i) Let $(a_{k})$ be a sequence in $\mathbb{R}$. If the series $\sum_{k=1}^{\infty}a_{k}$ of $(a_{k})$ converges, then its sum $\sum_{k=1}^{\infty}a_{k}$ is a real number.

(ii) Let $(a_{k})$ be a sequence in $[0,\infty]$, with partial sums $s_{n}=\sum_{k=1}^{n}a_{k}$ and sum $\sum_{k\in\mathbb{N}}a_{k}$, and let $t\in[-\infty,\infty]$. If $s_{n}\le t$ for every $n\in\mathbb{N}$, then

$$\sum_{k\in\mathbb{N}}a_{k}\le t.$$