I am confronted with the following argument which I think may not be right:
Let $(X_n)$ be a sequence of independent random variables s.t.
$P[X_n = 1] = 1- P[X_n = 0 ] = \frac{1}{n}$
in order to use Borell Cantelli we recall
$ X_n \to X \quad \text{ a.s. iff } \quad P[|X_n - X| > \epsilon \text{ i.o.}] = 0$
But $ P[|X_n - X| > \epsilon \text{ i.o.}] = 1$ by Borel Cantelli II as
$\sum^{\infty} P[X_n=1] =\sum^{\infty}{\frac{1}{n}} = \infty$
I m not firm enough yet with this material to make my argument solid why I think this may not be right. However I am wondering whether the $X_n$ are truely independent the way they are defined, i.e. can we actually use Borel Cantelli ?
Secondly, is it not the case, that $X_n$ converges pointwise to zero for every $\omega \in (0,1)$? Directly from the Definiton of convergence a.s. this would lead me to conclude, that $X_n \to 0$ a.s. right ? (I m probably assuming the standard Lebesgue measure on $[0,1)$ was implicit in the original argument that I find myself confronted with.)