If $X$ is a non-negative random variable , $r \in (0, \infty)$ then show that : $$\sum_{n=1}^{\infty} n^{r-1}P[X \geq n] \leq E[X^r] \leq 1 + \sum_{n=1}^{\infty} n^{r-1}P[X \geq n] $$
To show this , I want to use the fact that
$$E[X^r] = r \int_{0}^{\infty}x^{r-1}P[X>x] dx~.$$
But , I am unable to proceed .
Note : Proving the above inequalities can provide a criterion for finiteness of $r$-th order moment of a non-negative rv .
The inequality does not hold for all $r>0$. For example let $X=2$. The right hand inequality becomes $2^{r} \leq 1+ \sum\limits_{n=1}^{2} n^{r-1}=1+1+2^{r-1}$. But this is true only for $r \leq 2$.