Prove $(c-s)*P (Y \ge s) \ge EY − s$ for a random variable Y with y ∈ [0, c]

Question

Assume Y is a random value s.t. the RV takes value from the range [0, c] and for all s ∈ [0, c] prove the following: $$(c-s)*P (Y \ge s) \ge E(Y) − s$$

I tried to rewrite $P (Y \ge s)=1-\int f_ydy$ and then $EY=\int S_y dy$ Sy being the survivor function. Am I on the right track because I cant seem to progress here from here ( i tried multiple transformations but didnt workout.)

Answer 1

It is not difficult to verify that for every $y\in[0,c]$ we have:$$(c-s)1_{[s,\infty)}(y)\geq y-s$$

Since $Y\in[0,c]$ a.s. we are allowed to conclude that:$$(c-s)1_{[s,\infty)}(Y)\geq Y-s\quad \text{ a.s.}$$

From this it follows directly that:$$(c-s)\mathsf P(Y\geq s)=(c-s)\mathsf E1_{[s,\infty)}(Y)=\mathsf E(c-s)1_{[s,\infty)}(Y)\geq\mathsf E(Y-s)=\mathsf EY-s$$