Expectation of a non-negative random variable

Question

Let $X$ be a real non-negative random variable on the probability space $(\Omega,\mathcal{F}, \mathbb{P})$. Given that $$ E[X]=\int_\Omega \int_0^\infty \chi_{t<X}\,dt\,d\mathbb{P}=\int_0^\infty \mathbb{P}[ X\geq t]\,dt, $$ 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).

Answer 1

Notice that the summation in the middle is the Riemann sum of the decreasing function $f(t) = P(X>t).$ The monotonicity makes it easy to estimate the difference between the sum and the integral. Now, the question is: is the function Riemann integrable, and if not, does it matter?