Tricky sequence of random variables converges to $0$ with decreasing expectation

Question

I am studying for an exam and here is a sample question that I am unable to solve. I would appreciate your help in helping me prepare.

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Let $\{X_{i}\}_{i\in\mathbb{N}}$ be a sequence of random variables on a probability space $(\Omega, \mathcal{F}, P)$ such that $E(|X_{n}|) \leq 2^{-n}$ for each index $n$. Show $X_{i} \rightarrow 0$ almost surely as $n\to\infty$.

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So from the expectation bound, we have it getting smaller and smaller for each new variable in the sequence. I need to show the sequence converges almost surely, which, by what's on Wikipedia means I need to show that, $P(\lim_{n\to\infty} X_{n} = 0) = 1$. I'm not so sure how to relate the expectations to do this though.

I am also aware of chebyshev and other inequalities but I am still struggling to solve this problem.

Any help is appreciated

Answer 1

Use the Borel Cantelli lemma. In particular, in order to show that $X_i\to 0$ a.s., we show that for every $\varepsilon>0$, it is the case that $P(|X_n|>\varepsilon \quad \text{i.o})=0$ where i.o means "infinitely often". To this end note by Markov's inequality $$ \sum_n P(|X_n|>\varepsilon)\leq \sum_n \varepsilon ^{-1}E|X_n|\le \sum_n \varepsilon ^{-1}2^{-n}<\infty $$ whence $P(|X_n|>\varepsilon \quad \text{i.o})=0$ as desired.

Answer 2

$E\sum X_n^{2} =\sum EX_n^{2} <\infty$ and this implies $\sum X_n^{2} <\infty$ almost surely. In particular, the general term $X_n^{2} \to 0$ almost surely.