TITLE: On showing that if $f_n \to f, g_n \to g$ in $L^p$ then $max(f_n, g_n) \to max(f, g)$ in $L^p$
QUESTION [6 upvotes]: Let $(f_n)$ and $(g_n)$ be two sequences in $L^p(\Omega)$ with $1 \leq p < \infty$ such that $f_n \to f$ in $L^p(\Omega)$ and $g_n \to g$ in $L^p(\Omega)$. Let $h_n = max(f_n, g_n)$ and $h = max(f, g)$. Show that
$$h_n \to h \;\; \text{in} \;\; L^p(\Omega).$$
Can someone please give me a hint?
$f_n \to f$ in $L^p$ : $\|f_n - f\|_p \to 0$
$g_n \to g$ in $L^p$ : $\|g_n - g\|_p \to 0$.
How to show that $\|h_n - h\|_p \to 0$?
Thank you!

REPLY [6 votes]: Hint:
$$
\max(a,b)=\frac{a+b+|a-b|}{2},\quad a,b\in\mathbb{R}.
$$