You have
$$EX=\int_0^{\infty}xdF(x)$$
Notice that $dF(x)=d(1-F(x))$ and that $P(X>t)=1-F(t)$ and use integration by parts.
Now show that for monotone decreasing positive function
$$\sum_{n=0}^\infty f(n)\ge\int_0^{\infty} f(t) dt$$
Combine these two results and you get your desired result. Hint for the second, recall Riemman sums.