I have come up with 2 approaches but I'm not sure which one would be more appropriate.
1) $E[XY] = \sum_{m = 1}^{\infty} kP(X = k) * \sum_{n = 1}^{\infty} tP(Y=t)$
2) $E[XY] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xyf_{X,Y}(x,y)dxdy$
I am asked to prove the following.
$$E[XY] = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty} P(X \geq m, Y \geq n)$$