# Does this sequence converge almost surely or not?

I have a sequence of independent random variables $X_1, X_2,...$ such that $P(X_n = 1) = \frac{1}{n}$ and $P(X_n = 0) = 1 - \frac{1}{n}$. Using the second Borel-Cantelli Lemma, we have $\sum P(X_n \neq 0) = \sum P(X_n = 1) = \sum \frac{1}{n} = \infty$ implies that $X_n$ does not converge to 0 a.s.

But I feel like we can construct an example where almost convergence occurs. Let $\Omega = (0, 1)$, and let the probability measure be the Lebesgue measure restricted to $(0, 1)$. Finally, let $X_n = \mathbb{I}\{(0, \frac{1}{n})\}$. Then for any $\omega \in \Omega$, $\underset{n \to \infty}{\lim} X_n(\omega) = 0$, so $X_n$ converges to 0 a.s.

I don't understand why I'm getting contradicting conclusions. Any help would be much appreciated. Thank you!

The random variables $X_n := 1_{(0,1/n)}$ are not independent. This follows from the fact that
$$\lambda(X_n=1,X_m=1) = \lambda(X_m=1) = \frac{1}{m} \neq \frac{1}{n} \frac{1}{m} = \lambda(X_n=1) \lambda(X_m=1)$$
for any $m \leq n$. (Here $\lambda$ denotes the Lebesgue measure restricted to $(0,1)$.)