Proof for mean of non-negative discrete random variable

For a nonnegative integer-valued discrete random variable X, show that
E(X)=http://upload.wikimedia.org/math/3/d...c702e182f0.pngx=0 (to infinity) P(X>x). Can anyone help me solve this please?

So



This is a triangular region. So we can switch the order of the sum if we do it carefully.





Can you finish it?

Thank so much for your reply!
So to finish it then:

http://www.mathhelpforum.com/math-he...b874ace9-1.gif=http://upload.wikimedia.org/math/3/d...c702e182f0.pngx=0 (to infinity)xP(x)=E(x)?
Because x=0 --> y=1?

Well, there are no x's anymore. The x and y are just the summing variables.

The main observation at this point is that the sum:

almost is (whether it is x or y doesn't matter, it is just a summing variable). The only thing missing is the 0 term. The sum should go from 0 to , whereas this one starts at 1. But the 0 term is
.

So