Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits

In the setting of The Real Numbers: Standing Notation and Background, let $N$ be a natural number and let $[N]$ be the initial segment determined by $N$. All sums below are the finite sums of the field $\mathbb{R}$, formed from the map on $[N]$ indicated in the clause in question, and convergence of a sequence of real numbers is that of the indicated definition. Then the following hold.

1. (Nonnegativity)¶ Let $t:[N]\to\mathbb{R}$ satisfy $0\le t_{k}$ for every $k\in[N]$, and let $j\in[N]$. Then

If moreover $t_{k}=0$ for every $k\in[N]$, then $\sum_{k=1}^{j}t_{k}=0$.

2. (Each term is dominated by the sum)¶ Let $t:[N]\to\mathbb{R}$ satisfy $0\le t_{k}$ for every $k\in[N]$, let $j\in[N]$ and let $i\in[j]$. Then

3. (Limits)¶ For every $k\in[N]$ let $(a_{k,m})_{m\in\mathbb{N}}$ be a sequence of real numbers converging to the real number $A_{k}$, and let $j\in[N]$. Then the sequence whose $m$th term is $\sum_{k=1}^{j}a_{k,m}$ converges to $\sum_{k=1}^{j}A_{k}$.