The paragraph is given below:
But I do not understand the last statement, is it because by the definition of the $p-$norm if $\{f_{n}\} \rightarrow f$ then we have $$(\int_{E} |f_{n} - f|^p)^{1/p} \rightarrow 0 $$ then upon taking the $p^th$ power on both sides we get $$(\int_{E} |f_{n} - f|^p) \rightarrow 0. $$
Is my justification correct?
Of course $u_{n}^{p}\rightarrow u^{p}$ is equivalent to $u_{n}\rightarrow u$, here $u_{n},u\geq 0$. Such an equivalence follows by the continuities of the maps $a\rightarrow a^{p}$ and $b\rightarrow b^{1/p}$ for $a,b\geq 0$.