Consider $\{X_n\}$ that for some $p>0$, the sum $ \sum_{i=0}^{\infty}{E|X_n|^p} $ is finite. Show that $X_n\to 0$ almost surely.
Can someone help me find a way to solve this problem, or give a clue. cuz I have no idea how to solve it. maybe we should use Borel-Cantelli Lemma somehow.
Indeed, it is possible to solve this with the Borel-Cantelli lemma. Let $A_{n,\varepsilon}=\{\lvert X_n\rvert>\varepsilon\}$. Then by Markov's inequality, $$ \mathbb P\left(A_{n,\varepsilon}\right)=\mathbb P\left(\lvert X_n\rvert^p>\varepsilon^p\right)\leqslant \varepsilon^{-p}\mathbb E\left[\lvert X_n\rvert^p\right] $$ hence for each positive $\varepsilon$, the Borel-Cantelli lemma gives that $\limsup_{n\to\infty}\lvert X_n(\omega)\rvert \leqslant \varepsilon$ for each $\omega\in \Omega_\varepsilon$, where $\Omega_\varepsilon$ has probability one. To conclude, note that $X_n(\omega)\to 0$ on $\bigcap_{k\geqslant 1}\Omega_{1/k}$, which has probability one.
An other way to solve the problem is to note that the random variable $\sum_{n\geqslant 1}\lvert X_n\rvert^p$ is integrable (by Beppo-Levi lemma) hence almost surely finite hence $\lvert X_n\rvert^p\to 0$ almost surely.