If $\lim \int_0^1 |f_n(x)|=0$ and $|f_n|^2\leq g$ where g is integrable on [0,1], then $\lim \int_0^1 |f_n|^2=0$

Question

If $\lim \int_0^1 |f_n(x)|=0$ and $|f_n|^2\leq g$ where g is integrable, then $\lim \int_0^1 |f_n|^2=0$

Does anyone have any hints on how to begin? I think we have to use Hölder's or Dominated Convergence but I can't seem to see how. Also DCT requires point wise convergence which in the case need not be necessary. Any suggestions?

Answer 1

Suppose $\int |f_n|^{2}$ does not tend to $0$. There exists $\epsilon >0$ and a subsequence $(f_{n_k})$ such that $\int |f_{n_k}|^{2} \geq \epsilon$ for all $k$. Since $f_{n_k} \to 0$ in $L^{1}$ it has a subequence which converges to $0$ a.e. Now apply DCT to $(f_{n_k}^{2})$ to get a contradiction.