$\lim_{y\downarrow 0} yE(1/X ; X>y)=0$

Question

Let $X>0$ but do NOT assume $E(1/X)< \infty$. Show that $$\lim_{y\downarrow 0} yE(1/X ; X>y)=0$$.

I tried to use Jensen Inequality but this is not working.

There was another problem which was $$\lim_{y\rightarrow \infty} yE(1/X ; X>y)=0$$

It was easy since $\lim_{y\rightarrow \infty} yE(1/X ; X>y)< \lim_{y\rightarrow \infty} P(X>y)=0$. But this trick not working on the above problem.

Need Help!!

Answer 1

Hint: For any $\epsilon\gt0$, $$ \begin{align} y\operatorname{E}\left(\frac1X\,\middle|\,X\gt y\right) &=\int_{x\gt y}\frac yx\,\mathrm{d}\!\operatorname{P}(X\le x)\\ &=\int_{x\gt y}\frac yx\,\mathrm{d}\!\operatorname{P}(X\le x\land X\le\epsilon)\\ &+\int_{x\gt y}\frac yx\,\mathrm{d}\!\operatorname{P}(X\le x\land X\gt\epsilon)\\[3pt] &\le\operatorname{P}(y\lt X\le\epsilon)+\frac y\epsilon \end{align} $$ Note: this fails if $\operatorname{P}(X=0)\gt0$, however the assumption $X\gt0$ implies $\operatorname{P}(X=0)=0$.