Bolzano Weierstrass theorem in Lebesgue measure

Question

I need to check the correctness of this statement:

Let $f_n : \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of Lebesgue measurable functions. Is there a subsequence $\lim_{k\to \infty}f_{n_k}(x)$ that converges almost everywhere?

I guess the statement is wrong but not easy to construct the example.

Answer 1

The constant functions are measurable, so $f_n=n$ are measurable, but do not converge anywhere.