For $f\in C([\alpha,\beta],\mathbb{K})$ and $1\leq p\leq q\leq\infty$, prove $||f||_p\leq(\beta-\alpha)^{\frac{1}{p}-\frac{1}{q}}||f||_q$
This is an exercise from Amann's Analysis. It gives a hint that to use $||fg||\leq||f||_p||g||_{p'}$,where $\frac{1}{p}+\frac{1}{p'}=1$. But I don't know how to use it.