Fix $1 \leq p < \infty$. If $f_n \to f$ in $L^p(\mathbb R^d)$, $g_n \to g$ pointwise, and $\| g_n \|_{\infty} \leq M < \infty$ for all $n$, prove that $f_ng_n \to fg$ in $L^p(\mathbb{R}^d)$.
Clearly, what I want to do is bound $\|f_ng_n - fg \|_p$. My problem is how. I tried to split this difference using the Minkowski inequality, but then I get a lower bound, so that doesn't really help me. What should I bound this from above with? Another approach I was playing around with was to write $\| f_ng_n - fg \|_p \leq \|f_nM - fM \|_p = M\|f_n - f\|_p$, and proceed from there, but I'm not sure if this is actually accurate, and in particular how to justify it if it is accurate. Any help will be appreciated!
Not exactly, you need to separate like: $f_ng_n-fg=(f_n g_n-fg_n)+(fg_n-fg)$, for the first part, take $g_n$ out with the sup norm then $f_n \to f$ in $L_p$. For the second part, use dominated convergence theorem, when $g_n$ approaches $g$, $|fg_n-fg|<2M|f|$ which is the bound function.