Product of $L^p$-convergent sequences are Cauchy I'm working on showing:
If $\|f_n - f\|_p \to 0$ with $1≤p<\infty$ and $h_n \to h$ pointwise with $|h_n|<M$ for all $n$, then $\{f_nh_n\}$ is Cauchy in $L^p$
I've shown that $\|h_n-h\|_p \to 0$, but I'm having a hard time showing the product is Cauchy. I tried to manipulate the terms in the norm in additon to using Minkowski's inequality, but it hasn't worked thus far.
 A: Although $\|h_n - h\|_p$ may not converge to $0$, the dominated convergence theorem tells you that $\|f h_n  - fh\|_p \to 0$.  In particular $\{f h_n\}$ is Cauchy in the $L^p$ norm. Then apply the triangle inequality: for any $n$ and $m$ you have
\begin{align*} \|f_n h_n - f_m h_m\|_p &\le \|f_n h_n - f h_n\|_p + \|fh_n - fh_m\|_p + \|fh_m - f_m h_m\|_p \\
&\le K \|f_n - f\|_p + \|fh_n - fh_m\|_p + K \|f_m - f\|_p.
\end{align*}
All terms on the right tend to $0$ as $m,n \to \infty$.

On edit: you pointed out in the comments that you are actually trying to prove that $f_n h_n \to fh$ in $L^p$. You don't really need to go through Cauchy sequences to show this.  Using the fact that $\|fh_n - fh\|_p \to 0$ you have $$\|f_n h_n - fh\| \le \|f_n h_n - f h_n\|_p + \|fh_n - fh\|_p \le K \|f_n - f\|_p + \|fh_n - fh\|_p \to 0.$$
A: Being that $L_p$ is a complete space, there is $f\in L_p$ such that $\|f_n-f\|_p\xrightarrow{n\rightarrow\infty}0$.
$$ |f_nh_n -fh|\leq |(f_n-f)h_n| + |fh_n-fh|\leq |f_n-f|M+|f||h_n-h|
$$
As $|f||h_n-h|\leq 2M|f|$ and $|f||h_n-h|\xrightarrow{n\rightarrow\infty}0$ a.s., $\|f(h-h_n)\|_p\rightarrow0$ by dominated convergence.
Consequently
$f_nh_n$ converges to $fh$ in $L_p$ and so, $\{f_nh_n\}$ is a Cauchy sequence.
