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

Let the (Lebesgue-)measure of some $\displaystyle \Omega$ be finite and $\displaystyle 1 \le p \le q \le \infty$.

Then for all $\displaystyle u \in L^q(\Omega)$ it is also true that $\displaystyle u \in L^p(\Omega)$, whereby

$\displaystyle ||u||_p \le \text{meas}(\Omega)^{\frac{1}{p}-\frac{1}{q}}||u||_q$;

for $\displaystyle q=\infty$ set $\displaystyle \frac{1}{q}:=0$.

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

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