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Let $\left( X_n \right)$ be a sequence of independent random variables on the measure space $(\Omega, \xi,\mathbb{P})$ with $$ \mathbb{P} \left( X_n=1 \right)= p_n \text{ and } \ \mathbb{P} \left( X_n=0 \right)= 1- p_n. $$ I have proved that $$ X_n \to 0 \text{ in probability}\Leftrightarrow p_n \to 0 , \text{ as } \ n\to \infty $$ and $$ X_n \to 0 \text{ in the $L^p$ meaning of convergence} \Leftrightarrow p_n \to 0 , \text{ as } n \to \infty$$ My problem is to show the following implication: $$ X_n \to 0 \text{ almost surely} \Rightarrow \sum_{n=1}^{\infty} p_n < \infty. $$ I have some problem to make out what is the set $$ \{ w\in \Omega \ s.t. \ X_n(w) \nrightarrow 0 \} $$

Thanks in advance.

Notation: I'm using the notation and the definitions about convergence as in the book Probability by A.N. Shiryaev

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By the Borel Cantelli lemma, if $\sum_{n=1}^{\infty}p_n = \infty$ then

$$P(X_n = 1 \ \ \textrm{infinitely often}) = 1.$$

But clearly

$$\{X_n = 1 \ \ \textrm{infinitely often}\} \subseteq \{X_n \nrightarrow 0\}$$

So the latter set also has probability $1$. Hence $P(X_n \rightarrow 0) = 0$.

I have shown the contrapositive statement to your required result; the result is therefore proved.

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