User:Dfeuer/Generalized Sums of Positive Elements are Increasing

Theorem

Let $(G, +, \le)$ be an abelian totally ordered group.

Let $\{x_i : i \in I \}$ be an indexed set of (weakly) positive elements of $G$.

Let $J \subseteq I$.

Suppose that the generalized sums $\sum\{x_i: i\in I\}$ and $\sum\{x_j: j\in J\}$ both converge.

Then $\sum\{x_j:j \in J\} \le \sum\{x_i: i\in I\}$.

Proof

Follows from the fact that a generalized sum of positive elements converges to the supremum of the set of finite subsums and the fact that suprema are increasing.