measure theory - Corollary of Monotone Convergence Theorem

Let $(g_n)$ be a sequence in $M^+$, then

$\int \left(\sum_{n=1}^\infty g_n\right)d\mu=\sum_{n=1}^\infty\left(\int g_nd\mu \right)$

proof: Let $f_n=g_1+\cdots+g_n$, then $f_n$ is a monotone increasing sequence of functions in $M^+$. I wan t use Monotone Convergence Theorem but I don't know how to guarantee that $f_n$ converges to $f=\lim_{n\to \infty}\sum_{i=1}^n g_n=\sum_{i=1}^{\infty}g_n$.