Let us call a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ a norm if:
$f(v)\geq0$ for all $v\in\mathbb{R}^n.$
$f(v)=0\implies v=0.$
$f(v+w)\leq f(v)+f(w)$ for all $v,w\in\mathbb{R}^n.$
$f(\lambda v)=|\lambda| f(v)$ for all $v\in\mathbb{R}^n,\lambda\in\mathbb{R}.$
With this definition, is it true that if $v=(v_1,...,v_n)$, then $\frac{v_i}{f(v)}\leq1$ for all $i\in\{1,...,n\}?$ If possible, can you link a proof/counterexample?