Convergence of measurable functions by two conditions

I have the following task from my textbook:

Let $\left\{f_{n}\right\}_{n \in \mathbb{N}}$ be a sequence of measurable function on $M$ with $$ f_{n} \rightarrow f \text { a.s., } $$ where $f$ is also a measurable function. Here, a.s. means almost surely (= almost everywhere).

Show: if there exists a nonnegative measurable function $g$ satisfying the following conditions: $$ \left|f_{n}\right| \leq g \text { a.s. for all } n \in \mathbb{N} $$ and $$ \int_{M} g^{p} d \mu<\infty \text { for one } p>0, $$ then $$ \int_{M}\left|f_{n}-f\right|^{p} d \mu \rightarrow 0 \text { for } n \rightarrow \infty . $$

These notations remind me a little bit of the Hölder inequality.

What I have got:

I thought that we have $$ |f_n|^p \leq g^p a.s. $$ Thus $$ \int_M |f_n|^p d\mu \leq \int_M g^p d\mu \leq infty $$

And as $$ f_n \rightarrow f $$ then the rest follows?

For $p>1,$ the function $x\mapsto x^p$ is convex so $|f_n-f|^p=2^p\left|\frac{f_n}{2}+\frac{(-f)}{2}\right|^p\leq 2^{p-1}(|f_n|^p+|f|^p)$ so $$ 2^{p-1}(|f_n|^p+|f|^p)-|f_n-f|^p\geq 0. $$ In the standard way that dominated convergence theorem is proven, just apply Fatou's lemma.