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!