I want to prove
$D$ is a measurable set and $m(D)<\infty$. Two sequences of functions $\{f_n\}_{n\geq1}$ and$\{f_n\}_{n\geq1}$ are defined on $D$. $f_n$ converges to $f$ in measure and $g_n$ converges to $g$ in measure. Prove $f_ng_n$ converges to $fg$.
In the solution manual, the last step is that
Since the subsubsequecne $f_{n_{k_i}}g_{n_{k_i}} $ converges to $fg$ in measure, $f_ng_n$ converges to $fg$ in measure.
But I am not sure if this is correct.
By the way, the whole proof is
$\exists~\text{a subsequence}~ \{f_{n_k}\}$ of $\{f_n\}$ s.t. $f_{n_k}\rightarrow f$ a.e.,and $\exists~\text{a subsequence}~ \{g_{n_{k_i}}\}$ of $\{g_{n_k}\}$ s.t. $g_{n_{k_i}}\rightarrow g$ a.e. Then, $f_{n_{k_i}}g_{n_{k_i}} \rightarrow fg$ a.e.==> $f_{n_{k_i}}g_{n_{k_i}} \rightarrow fg$ in measure==>$f_ng_n \rightarrow fg$ in measure.