TITLE: Is $|x-y|^n\leq 2^n(|x|^n+|y|^n)$?
QUESTION [5 upvotes]: Is 
$$|x-y|^n\leq 2^n(|x|^n+|y|^n)$$
for all $x,y\in\mathbb{R}$ and $n=1,2,3,\dots$ a standard inequality? If so, what's its name or how do you prove it?

REPLY [7 votes]: $|x|=(|x|^n)^{\frac{1}{n}}\le (|x|^n+|y|^n)^{\frac{1}{n}}$ and $|y|=(|y|^n)^{\frac{1}{n}}\le (|x|^n+|y|^n)^{\frac{1}{n}}$. So
$$|x-y|\leq |x|+|y|\leq 2(|x|^n+|y|^n)^{\frac{1}{n}}$$ 
and consequently $|x-y|^n\leq 2^n(|x|^n+|y|^n)$