Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits

Let $\mathbb{R}$ be the ordered field of real numbers, with order $\le$ and absolute value $|\cdot|$, natural numbers being read in $\mathbb{R}$ through the canonical map. Let $G$ be a nonempty finite set and let $h,h':G\to\mathbb{R}$ be maps. Sums over a finite index set are those of Sum over a Finite Index Set, and for a subset $E\subseteq G$ the sum $\sum_{x\in E}h(x)$ is that of the restriction of $h$ to $E$. Then the following hold.

1. (Comparison)(i) If $h(x)\le h'(x)$ for every $x\in G$, then

$$\sum_{x\in G}h(x)\le\sum_{x\in G}h'(x).$$

2. (Nonnegativity)(ii) If $0\le h(x)$ for every $x\in G$, then $0\le\sum_{x\in G}h(x)$.

3. (Monotonicity in the index set)(iii) Suppose $0\le h(x)$ for every $x\in G$. Then for every nonempty $E\subseteq G$,

$$\sum_{x\in E}h(x)\le\sum_{x\in G}h(x).$$

4. (Each term is dominated by the sum)(iv) Suppose $0\le h(x)$ for every $x\in G$. Then $h(y)\le\sum_{x\in G}h(x)$ for every $y\in G$.

5. (Absolute value)(v)

$$\Bigl|\sum_{x\in G}h(x)\Bigr|\le\sum_{x\in G}|h(x)|.$$

6. (Counting)(vi) If $G$ has $p$ elements, then $\sum_{x\in G}1=p$, the left-hand side being the sum over $G$ of the map with constant value $1$.

7. (Limits)(vii) For every $x\in G$ let $(a_{x,m})_{m\in\mathbb{N}}$ be a sequence of real numbers that converges to a real number $A_{x}$. Then the sequence whose $m$th term is $\sum_{x\in G}a_{x,m}$ converges to $\sum_{x\in G}A_{x}$.