Can anyone give me a hint for proving the following:

Let $\Omega$ be a measure space. Assume $f \in L^p(\Omega)$ and $g \in L^q(\Omega)$ with $1 \leq p, q \leq \infty$ and $\frac1p + \frac1q \leq 1$. Prove that $fg \in L^r(\Omega)$ with $\frac1r = \frac1p + \frac1q$. 

Note: One should be able to use (the standard) Hölder inequality. Notice that if you have $\frac1p + \frac1q = 1$ you recover the former result.