How to prove this function is $L^{r}(E)$

Question

Let $p,q\in[1,\infty)$, $E\subseteq\mathbb{R}$ measurable with finite measure. Let $u\in L^{p}(E)\cap L^{q}(E)$, fix $\alpha\in[0,1]$ and define: $$\frac{1}{r}=\frac{1-\alpha}{p}+\frac{\alpha}{q}$$ Show $u\in L^{r}(E)$ and: $$\|u\|_{L^{r}(E)}\leq\|u\|^{1-\alpha}_{L^{p}(E)}+\|u\|^{\alpha}_{L^{q}(E)}$$

Answer 1

See the interpolation section in https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality.