Suppose that $f_k$ converges to $f$ in $L^p$ where $1\leq p < \infty$ and $g_k$ converges pointwisely to $g$ with $\|g_k\|_\infty<M<\infty$ for all $k$. Prove that $f_kg_k$ converges to $fg$ in $L^p$.
My attempt: If $h_k=|f_kg_k|^p$, use the fact that $g_k$ converges pointwisely, $h_k$ converges to $|fg|^p$ a.e. And then, $h_k=|f_kg_k|^p<M^p|f_k|^p$ which is in $L^1$ for all $k$. Now I would like to apply the general version of the Lebesgue Dominated Theorem to conclude that $\int h_k$ converges to $\int |fg|^p$ since $\int M^p|f_k|^p$ converges to $\int M^p|f|^p$. But one of the hypothesis is that $M^p|f_k|^p$ needs to converge to $M^p|f|^p$ a.e. And I don't know if that is true.
Is my argument wrong, or is there any better way to prove it?
$M^p|f_k|^p$ needn't converge to $M^p|f|^p$ a.e, but Riesz-Fischer tells us there is a subsequence which does, you can use this to your advantage with the following lemma:
In a normed linear space, if $\{x_n\}$ is a Cauchy sequence, and $\{x_{n_k}\}$ converges to $x$, then so does the original sequence. Use your argument to show $\{g_kf_{n_k}\}$ converges to $fg$ in $L^p$, and also that $\{g_kf_{k}\}$ is Cauch, and your'e done!