TITLE: Solution to Billinglsey (1995) problem 20.22
QUESTION [1 upvotes]: Let $Y_1\leq Y_2\leq ...$ be random variables s.t. $\mathrm{plim} Y_n = Y$. Show that $Y_n \to Y$ with probability 1.
Some hints?
My strategy would be to prove that $\sum P(\lvert Y_n -Y \rvert > \epsilon) < \infty$ and then use the first Borel-Cantelli lemma. But I am not able to prove the former. Other ways are of course welcome.

REPLY [2 votes]: Since $Y_n\to Y$ in probability, you can extract a subsequence $Y_{n_k}$ that converges a.s. to $Y$.  Can you use this fact to complete the argument?
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