Suppose $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is strictly increasing, continuous function and $f_n$ is sequence of measurable function. Prove that if $\phi(f_n)$ is Cauchy in $L^1$, then there exists a subsequence $f_{n_i}$ converges almost everywhere.
How can I prove it? I can only get a subsequence of $\phi(f_{n_i})$ converges almost everywhere.