combinatorics - Double summation limits

Is there a way to "see" that

$$\sum\limits_{r=0}^\infty \sum\limits_{x=r+1}^\infty \mathbb P(X=x)=\sum\limits_{x=1}^\infty\sum\limits_{r=0}^{x-1}\mathbb P(X=x)\; ?$$ Thanks.

Answer

It doesn't matter what you sum (as long as the sums are convergent). The points in the $(r,x)$ plane that are being summed over can be illustrated by a diagram like this:

x

5 *****
4 ****
3 ***
2 **
1 *

  012345  r

The two sides in the identity correspond to summing by columns first or rows first.