$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
Since
$$ \sum_{n=1}^{\infty} \left| f_n - f \right| $$
is integrable, this sum is finite a.e. This implies that the series converges a.e., hence we have $|f_n - f| \to 0$ a.e.