## Equivalence of Norms

I'm somewhat stuck with the simple proof of the following:

Let the (Lebesgue-)measure of some $\Omega$ be finite and $1 \le p \le q \le \infty$.
Then for all $u \in L^q(\Omega)$ it is also true that $u \in L^p(\Omega)$, whereby
$||u||_p \le \text{meas}(\Omega)^{\frac{1}{p}-\frac{1}{q}}||u||_q$;
for $q=\infty$ set $\frac{1}{q}:=0$.

Proof: If $q=\infty$, then $||u||^p_p=\int_{\Omega}|u(x)|^p\,dx \le \text{meas}(\Omega) \sup_{\Omega \setminus N}|u|^p$, where N ist a large enough null set.

If $q<\infty$, then the Hölder inequality should help, but I'm somewhat confused by the suitable choice of exponents.