I'm required to show that if $\displaystyle X\geq 0$ then $\displaystyle \sum_{n=1}^{\infty}\mathbb{P}(X\geq n) \leq \mathbb{E}[X] \leq 1 + \sum_{n=1}^{\infty}\mathbb{P}(X\geq n)$

For the continuous case it was easy enough to express $\displaystyle \mathbb{E}[X]$ in terms of the CDF and the result followed from there...

But for the discrete case I'm a little stuck as to how to start...

$\displaystyle \mathbb{E}[X] = \sum_{i=0}^{m}x_iP(X=x_i)$

$\displaystyle \sum_{n=1}^\infty\mathbb{P}(X\geq n) = \sum_{n=1}^\infty\sum_{i=n}^m\mathbb{P}(X=x_i)$

hmmmm....

ay help would be much appreciated