probability theory - Finite expectation of a random variable

$X \geq 0$ be a random variable defined on $(\Omega,\mathcal{F},P)$. Show that $\mathbb{E}[X]<\infty \iff \Sigma_{n=1}^\infty P(X>n) < \infty$.

I got the reverse direction but I am struggling with the $"\implies"$ direction. So far, I have the following worked out:

$\mathbb{E}[X]<\infty$

$\implies \int_0^\infty (1-F(x)) dx < \infty$ (where $F$ is the distribution function of the random variable X)

$\implies \int_0^\infty (1-P(X\leq x)) dx < \infty$

$\implies \int_0^\infty P(X>x) dx < \infty$

Consider $\int_0^\infty P(X>x) dx$

$= \Sigma_{n=1}^\infty \int_{n-1}^n P(X>x) dx$

This is the point I am stuck at. Any help will be deeply appreciated!