TITLE: Another question on almost sure and convergence in probability
QUESTION [18 upvotes]: Convergence in probability implies convergence on a subsequence almost surely.
But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we pick does not depend on the $\omega$ right?

REPLY [4 votes]: Also, you can directly apply Borel Cantelli to the sequence of events $|X_{n_k}-X|>2^{-k}$.