- #1

- 274

- 1

## Homework Statement

Let [itex]\{a_{n,k}:n,k\in\mathbb{N}\}\subseteq[0,\infty)[/itex]. Prove that [itex]\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}a_{n,k}=\sum\limits_{k=0}^{\infty}\sum\limits_{n=k}^{\infty}a_{n,k}[/itex].

## Homework Equations

## The Attempt at a Solution

I am pretty certain that the claim is true because when I expand them out I get

[itex]\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}a_{n,k}=a_{0,0}+a_{1,0}+a_{1,1}+a_{2,0}+a_{2,1}+a_{2,2}+...[/itex] and

[itex]\sum\limits_{k=0}^{\infty}\sum\limits_{n=k}^{\infty}a_{n,k}=a_{0,0}+a_{1,0}+a_{2,0}+...+a_{1,1}+a_{2,1}+...[/itex]

which look to me like reorderings of each other. The problem is I am not sure about how I should approach proving that they are in fact equal.