$L^p$ convergence from pointwise convergence and the convergence of $L^p$-norm

Let $f,\lbrace f_k\rbrace\in L^p$. I want to prove that if $f_k\to f$ a.e. and $\|f_k\|_p\to \|f\|_p$ then $\|f-f_k\|_p\to 0$.

I want to use some tricky Holder-inequality to use $\|f_k-f\|_1\to 0$ as $k\to \infty$ but I couldn't use the condition $\|f_k\|_p\to \|f\|_p$.

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merged by Willie Wong♦Jul 16 '11 at 1:49

this question was merged with Convergence a.e. and of norms implies that in Lebesgue space because it is an exact duplicate of that question.