Almost everywhere convergence and $L^p$ probability

Question

Let $(\Omega, \mathcal F, P)$ be a probability space. If $X_n \leq C$ for each $n\geq 1$ and $X_n \to X$ almost everywhere, prove that $X_n \to X$ in $L^p$.

So far I have just showed that $X, X_n \in L^p$. For the $L^p$ convergence I have just used the DCT to write $\int_{\Omega}X_n dP \to \int_{\Omega}X dP$, but don't know what else to do.

Answer 1

Hint:

As stated by the comment below the question, it is not true unless you have $|X_n| \le C$ instead.

If we assume that, apply the DCT to $$\int_\Omega |X_n-X|^p dP$$