fa.functional analysis - A question on weak derivative - Sobolev spaces

Let $Omega$ be an open set in $R^n$, and $f in L^1_{loc}(Omega)$, such that foreach multiindex $alphain N^n$, $|alpha| = l$ f has weak derivative $D^alpha f$ in $L^p(Omega)$, with $1leq pleq infty$.

In general it is not true that $fin L^p(Omega)$, but it have to be true that $fin L^p_{loc}(Omega)$. How can be shown that?