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

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!

Answer 1

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