If $f_k\rightarrow f$ in $L^p$, $1\leq p < \infty$, $g_k\rightarrow g$ pointwise, and $\|g_k\|_{\infty} \leq M$ for all $k$. Prove that $f_k g_k\rightarrow fg$ in $L^p$.
Attempted proof - Let $\{f_k\}$ be a sequence in $L^p$ that converges to $f\in L^p$. Then for $\epsilon > 0$ there exists $N\in\mathbb{N}$ such that $$\|f_n - f\| < \frac{\epsilon}{2} \ \forall n\geq N$$ Now suppose we have another sequence $\{g_k\}$ that converges to $g$ pointwise and $\|g_k\|_{\infty} \leq M$. Now by Minkowski's inequality, we have $$\|f_k g_k - fg\|_{p} \leq \|f_k g_k - fg_k\|_{p} + \|fg_k - fg\|_{p}\leq \|g_k(f_k - f)\|_{p} + \|f(g_k - g)\|_{p}$$
Not really sure to go from here any suggestions is greatly appreciated.
Use the useful inequality $$\left\|f\cdot g\right\|_p\leq\left\|f\right\|_p\cdot\left\|g\right\|_{\infty}$$