Prove\Refute: for every norm in $\mathbb{R}^n: \left \| x \right \|\leq \max (\left \| x+y \right \|,\left \| x-y \right \|)$ I need to prove or refute that for every norm in $\mathbb{R}^n$:$ \left \| x \right \|\leq \max (\left \| x+y \right \|,\left \| x-y \right \|)$. It's been quite a while since I studied Linear algebra 1. I tried to look for vectors $x$ and $y$ such that they will refute the claim, but I didn't find any, so I tried to prove the question by showing the explicit sum of each norm, and go on form that, but it didn't do either.
(sorry if the question is too easy or silly)
Any help?
Thank you very much!

A geometric way to think of this is to note that balls in normed spaces are convex, and $x$ lies on the line segment between $x+y$ and $x-y$.  If $\|x\|$ were greater than $\|x-y\|$ and $\|x+y\|$, then $\{z:\|z\|<\|x\|\}$ would be an open ball that contains $x-y$ and $x+y$, but not their midpoint $x$, and therefore it would not be convex.
