Quick note before: We define $W^{1,p} = u\in L^p(I); g\in L^p(I)$ such that $\int_{I}u\varphi' = -\int_{I}g\varphi, \forall \varphi \in C_c^\infty(I)$
I is an open interval $(a, b)$ and $p\in \mathbb{R}$ with $1\leq p\leq \infty$
In Brezis' Functional Analysis, PDE, and Sobolev Spaces it says that $W^{1,p}$ is equipped with the norm $||u||_{L^p} + ||u'||_{L^p}$ or sometimes if $1 < p < \infty$ with the equivalent norm $\big(||u||_{L^p}^p + ||u'||_{L^p}^p\big)^\frac{1}{p}$.
How can I show that the two norms are equivalent? I know the definition for equivalent norms $\alpha||x||_X\leq ||x||_1 \leq \beta||x||_X, \forall x \in X$ but for some reason I'm just incappable to show that. Thanks in advance