Let $X_1, \ldots, X_n, \ldots$ be a sequence of Bernoulli random variables, $X_k \sim Bern(p_k)$. Prove that
$$
X_n \xrightarrow{a.s.} 0
$$
if and only if
$$
\sum_{k = 0}^{+\infty} p_k < +\infty.
$$
The "if" part is an easy implication from the Borel-Cantelli theorem, but I do not have any ideas about the second part of the problem (why convergence a.s. implies the convergence of the series)
A.s. convergence of Bernoulli sequence
almost-everywhereborel-cantelli-lemmasprobability
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Best Answer
The converse is not true. Take $(\Omega,\mathcal F,\mathbb P)=\big((0,1),\mathcal B(0,1),m\big)$ where $m$ is the Lebesgue measure on $(0,1)$. Let $$X_n(\omega ):=\boldsymbol 1_{\left(0,\frac{1}{n}\right)}(\omega ).$$ Set $$p_n:=\mathbb P\{X_n=1\}=\frac{1}{n}.$$ Then, $X_n\sim \text{Bern}(p_n)$ and $X_n\to 0$ a.s. However, $$\sum_{n\in\mathbb N}p_n=\sum_{n\in\mathbb N}\frac{1}{n}=\infty .$$