# $X_1, X_2, \ldots$ are independent, $X_n \longrightarrow 0$ almost surely. Prove that $\sum_{n=1}^{\infty} P \left( |X_n > 1| \right)$ is convergent

Random variables $$X_1, X_2, \ldots$$ are independent, $$X_n > \longrightarrow_{n\longrightarrow \infty} 0$$ almost surely. Prove that $$\sum_{n=1}^{\infty} P \left( |X_n > 1| \right)$$ is convergent.

Assuming the contrary that the series diverges, I am using the second Borel-Cantelli lemma as given on Wikipedia:

if $$\sum_{n=1}^{\infty} P \left( |X_n > 1| \right)$$ is divergent, and the events $$\{ \omega : |X_n(\omega)|>1 \}_{n=1}^{\infty}$$ are independent (follows from the statement of the problem),

then $$P \left( \lim_{n \longrightarrow \infty} \sup \{ \omega : |X_n(\omega)|>1 \} \right) = 1,$$ which contradicts given condition "$$X_n \longrightarrow 0$$ almost surely".

Is my solution correct?

• Yes, it is correct. Nov 23, 2023 at 11:24

This is correct. The only suggestion is to write $$\limsup_{n\to\infty}$$ instead of $$\lim_{n\to\infty}\sup$$.