>Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$ Does the $X_n\underset{a.s}{\to}0$

Question

Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$
Does the $X_n\underset{a.s}{\to}0$

I tried to use Borel-Cantelli but I get that $\sum_{i=1}^{\infty}\mathbb{P}(A_n^{\epsilon})=\infty$
where $A_n^{\epsilon}=\{|X_n-X|\geq\epsilon\}$
any hint please

Answer 1

$\sum P(X_n=1)=\sum \frac 1 n =\infty$. By independence and Borel Cantelli Lemma this implies $P(X_n =1 i.o.) =1$. Hence it is not true that $X_n \to 0$ almost surely.