How to show formally that $\mathbb{E}[X|X\geq c]\geq c$?

Question

I know that this is too obvious, but I've been trying to prove this formally but I can't. Intuitively, we 'know' that $X$ is only taking values which are equal or larger than $c$, so the average of all those values will be at least $c$ or greater.

Answer 1

If $\mathbb{P}[X>c]>0$ then, following the advice of @KaviRamaMurthy $\mathbb{E}[X 1_{X>c}]\geq c\mathbb{P}[X>c]$

So it follows that $\mathbb{E}[X|X>c]=\frac{\mathbb{E}[X 1_{X>c}]} {\mathbb{P}[X>c]}\geq c$

Thanks a lot!