A question about dominated convergence theorem

It is easier to argue by contradiction. Assume the contrary that

$$\int_0^1 |f_n|^2 dx$$

does not converge to $0$. Then by picking a subsequence if necessary, assume that there is $\epsilon_0>0$ so that

$$\tag{1} \int_0^1 |f_n|^2 dx \ge \epsilon_0.$$

The fact that $\int_0^1 |f_n| dx \to 0$ implies that (by picking a subsequence if necessary) $f_n \to 0$ almost everywhere. Thus $|f_n|^2 \to 0$ almost everywhere. Now one can use the condition $|f_n|^2 \le g$ and Lebesgue's dominated convergence theorem to conclude

$$ \lim_{n\to\infty} \int_0^1 |f_n|^2 dx = \int_0^1 \lim_{n\to \infty} 0 dx = 0.$$

But this contradicts (1).