Absolutely continuous function in $\mathcal{L}_p$

Question

Let $u$ be an absolutely continuous function and $u'\in\mathcal{L}_p(I)$, where $I$ is bounded.

How to prove that $u\in\mathcal{L}_p$??

I tried to write $u$ as

$$u(x)=u(x_0)+\int_{x_0}^xu'(t)dt$$

and then I take the norm for both sides alongside with triangle inequality but this could not help me at all.

I Appreciate any help.

Answer 1

Since $u'$ is $L^p$ on a finite interval it is $L^1$ so $u$ is actually bounded. (I assume $p \geq 1$ here as usual.) And again the interval is finite so that makes $u$ also be $L^p$.