My task is to show that on a finite measure space $(X,A)$, if $f_n \to f$ in measure, then every subsequence of $\{f_n\}$ has a subsequence that converges to $f$ almost everywhere.
I believe I was successful in showing the converse, but one thing confuses me:
I saw proof in my textbook that if $f_n \to f$ in measure, then there is a subsequence that converges to $f$ a.e.
However my problem asks that every subsequence of $f_n$ have a convergant subsequence.
I think I'm getting myself confused after thinking of subsequences of subsequences for too long, and a fresh take on things would be very helpful.
I think I remember seeing someone say that it should be a consequence of Egorov's Theorem, but I don't see how that is since it assumes pointwise convergence a.e. so how could it be used to prove it?