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Here's the problem transcribed from the book:

Use the Borel Cantelli lemma to prove that given any sequence of random variables $\{X_n, n \ge 1\}$ whose range is the real line, there exist constants $c_n \to \infty$ such that $$ P[\lim_{n \to \infty} \frac{X_n}{c_n}=0] = 1. $$

Give a careful description of how you choose $c_n$.

Basically I get confused when I think about picking a $c_n$ such that $\sum P(\frac{X_n}{c_n} \ge \epsilon) < \infty$.

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  • $\begingroup$ There seems to be a typo. Is it X_n/c_n instead of a_n/b_n ? $\endgroup$ – madprob Oct 2 '12 at 1:46
  • $\begingroup$ right, that's true. sorry $\endgroup$ – Taylor Oct 2 '12 at 1:48
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Do you know a set of constants $b_n$ such that $P(|X_n| > b_n) \leq \frac{1}{n^2}$?

Think about the quantiles of $|X_n|$.

Now, can you choose $c_n$ s.t. $P(\frac{|X_n|}{|c_n|} > \epsilon) = P(|X_n| > b_n)$?

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