How to prove $\lim\limits_{k\to \infty}f_{n_k}(x)=0$ a.e. in $[0,1]$

Question

$\{f_n(x)\}_{n>0}$ is Lebesgue measurable function on $[0,1]$ and there is a sequence $t_n$. $\sum\limits_{n=1}^\infty |t_n|=\infty$ s.t. $\sum\limits_{n=1}^\infty |t_nf_n|<\infty$ how to prove there is a subsequence $\{f_{n_k}(x)\}_{k>0}$ of $\{f_n(x)\}_{n>0}$ s.t. $\lim\limits_{k\to\infty}f_{n_k}(x)=0$ a.e. in $[0,1]$