probability - Expectation of a non-negative random variable

Let $X$ be a real non-negative random variable on the probability space $(\Omega,\mathcal{F}, \mathbb{P})$. Given that
$$
E[X]=\int_\Omega \int_0^\infty \chi_{t $$show that, for all $\epsilon>0$$$
E[X]\leq \sum_{n=0}^\infty \epsilon\mathbb{P}[X\geq n\epsilon]\leq E[X]+\epsilon.
$$
Tried to use some Fubini combined with rewriting stuff as countable sums (like $\mathbb{P}[X\geq t]=\mathbb{P}\left[\bigcup_{n=1}^\infty \{X\geq nt\}\right]$) but I am a bit lost. Some intuition is also highly appreciated (I think I am beaten by the misunderstanding of notation).