Nonnegativity and Monotonicity of a Sum over a Finite Index Set

In the setting of The Real Numbers: Standing Notation and Background, 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)¶ If $h(x)\le h'(x)$ for every $x\in G$, then

2. (Nonnegativity)¶ 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)¶ Suppose $0\le h(x)$ for every $x\in G$. Then for every nonempty $E\subseteq G$,