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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?

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    $\begingroup$ Yes, it is correct. $\endgroup$ Nov 23, 2023 at 11:24

1 Answer 1

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

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