almost surely convergence of a random variable

Question

I came across the following problem: Let $\xi\in L^{2}(\mathbb{P})$. Can we obtain that $\xi/T^{\epsilon}$ converges almost-sure to zero, for any $\epsilon$ as $T$ tends to infinite? Thus, is $\frac{\xi}{T^{\epsilon}}\overset{a.s.}{\rightarrow }0$ right? Thanks for your consideration.

Answer 1

$X=\lim_{T\to\infty}\xi/T^{\epsilon}$ exists and by the dominated convergence theorem $$0=\lim_{T\to\infty}E(\lvert\xi/T^{\epsilon}\rvert)=E(\lvert X\rvert)$$ so $X=0$ almost surely.