Partial Sums of a Sequence in [0,∞]: First Term, Recursion and Two-Term Sums

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $[-\infty,\infty]$, $[0,\infty]$ and the order $\le$ of $[-\infty,\infty]$ be as in The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), and the sum $+$ on $[0,\infty]$ as in Addition and Multiplication on [0,∞] (i). Let $(a_{k})$ be a sequence in $[0,\infty]$, let $s_{n}=\sum_{k=1}^{n}a_{k}$, for $n\in\mathbb{N}$, be its partial sums, and let $\sum_{k\in\mathbb{N}}a_{k}$ be its sum.

(i) $s_{1}=a_{1}$.

(ii) $s_{n+1}=s_{n}+a_{n+1}$ for every $n\in\mathbb{N}$.

(iii) If $a_{k}=0$ for every $k\in\mathbb{N}$ with $k\neq1$ and $k\neq2$, then $\sum_{k\in\mathbb{N}}a_{k}=a_{1}+a_{2}$.