Pointwise convergence implies $L^p$ Simply, why is it that convergence pointwise, $u_j \rightarrow u$, implies convergence in $L^p$ if $|u_j(x)| \le g(x)$ for some $g$ in $L_+^p$?

Simply, Lebesgue's dominated convergence theorem. "Domination" by $g$ is necessary: a traveling square wave $f_n=\chi_{[n,n+1]}$ converges pointwise to zero but has norm 1 for all $p$. 
