Theorem: If $1 \le p \le q \le \infty $, then $L^q (\Omega) \subset L^p_{loc} (\Omega) $.
My attempt: I proved that $L^q (\Omega) = \{ f: \Omega \rightarrow R$ measurable such that $f\big|_{\Omega'} \in L^q(\Omega'), \forall$ $ open$ $ \Omega' \subset \subset \Omega$}. ($\Omega' \subset \subset \Omega$ means that $\Omega' \subset \Omega$ and $\bar{ \Omega'}$ is compact). I don't know what to do next. (I don't know if what I proved is useful to prove the theorem).