I found this inequality in a paper and wondered if it is really true because I'm struggling to prove it:
For any random variable $Y$ and value $a \in \mathbb{R}$ we have: $$\mathbb{E}[Y] \leq a + \int_a^\infty \mathbb{P}(Y \geq t)dt$$
I tried expanding $\mathbb{E}$, but so far no luck. Maybe there is just a special case in which it holds, but I didn't find anything on Google. Help would be greatly appreciated.