I'm having a little trouble with this homework problem:
Suppose $\mu(X)<\infty$, $f_n\in L^1$, $f_n\to f$ a.e., and there exists $p>1$ and a constant $C>0$ such that $$\|f_n\|_p\leq C$$ for all $n.$ Prove that $f_n\to f$ in $L^p$.
Here's what I've done so far.
Without loss of generality, suppose that $\mu(X)=1$. Since $\|f_n\|_p\leq C$, we have that $f_n\in L^p$ for all $n$. Furthermore, since $|f_n|^p\to |f|^p$, we have that $$\int |f|^p\leq \liminf \int|f_n|^p\leq C^p$$ by Fatou's lemma, so $f\in L^p$ as well.
Since $|f_n- f|\to 0$, we have that $|f_n-f|^p\to 0$.
I am thinking of applying the Dominated convergence theorem to $|f_n-f|^p$, but I cannot think of a bound.