$n^{\alpha}U_{(1)} \to 0$ almost surely for any $\alpha < 0.5$

Question

Let $U_i$ be a sequence of identically distributed independent random variables with following density function $f(x) = 2x \ \ x \in[0,1]$ and $0$ everywhere else. I need to show $n^{\alpha}U_{(1)} \to 0$ almost surely for any $\alpha < 0.5$.

$U_{(1)} = min(U_1,U_2,\dots,U_n)$

My attempt: I have tried to use Borel-Cantelli lemma and got the following series

$\sum_{n=1}^{\infty}P(n^\alpha U_{(1)} \geq \epsilon) = \sum (1-\frac{\epsilon^2}{n^{2\alpha}})^n $

If I show the above series converges, I couldn't prove above series converges.

Answer 1

Hint: $1-t \leq e^{-t}$ for any $t \geq 0$ $\sum e^{-cn^{s}}<\infty$ for any $c,s >0$. [For any positive integer $k$ we have $e^{-cn^{s}} \leq \frac 1{(cn^{s})^{k}} {k!}$ choose $k$ such that $sk>1$].