True or false: $x\neq y$ in a normed vector space, then $\Vert x\Vert \neq \Vert y\Vert$

Question

Let $(\Vert\cdot\Vert,X)$ be a normed vector space. Let $x,y\in X$ be given.

Claim: If $x\neq y$, then $\Vert x\Vert \neq \Vert y\Vert$.

This is not true in general, because, for instance putting $y=-x$, then $\Vert y\Vert = \Vert -x\Vert=\Vert x\Vert$. Is my reasoning correct?

Answer 1

You are almost there! Based on the proposed counter example, you should also consider that $x\neq 0$.