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.