Let $X_{n}$ be a sequence of i.i.d. random variables with $P(X_n = -1) = P(X_n = 1) = \frac12$.

Prove that: $$X = \lim_{n\to\infty} \sum_{k = 1}^{n} e^{-k} X_{k}$$ exists almost surely.

I'm not so sure how to approach this problem. I am studying for an exam, and I would like to learn how to solve this problem. I have tried to use the Borel-Cantelli lemma by defining a sequence of events and showing that the sum is less than infinity. For example, I know $E[X_{n}] = 0$, so if I can write an expectation as the sum of probabilities, then this sum would go to $0$.

Can someone please help me?


$|X_k|=1$ almost surely so the series $\sum\limits_{k=1}^{\infty} e^{-k} X_k$ is absolutely convergent almost surely.


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