scaling of the open ball in a normed vector space

Let $V$ be a vector space over a field $F$ (real or complex), and let $\\|\\cdot\\|$ be a norm on $V$ . Further,for $r>0$ , $v\\in V$ , let

B_{r}(v)=\\{w\\in V:\\|w-v\\|<r\\}.

Then for any non-zero $\\lambda\\in F$ , we have

\\lambda B_{r}(v)=B_{|\\lambda|r}(\\lambda v).

The claim is clear for $\\lambda=0$ , so we can assume that $\\lambda\eq 0$ .Then

$\\displaystyle\\lambda B_{r}(v)$	$\\displaystyle=$	$\\displaystyle\\{z\\in V:\\\|w-v\\\|<r\\ \\mbox{and}\\ z=\\lambda w\\}$
	$\\displaystyle=$	$\\displaystyle\\{z\\in V:\\\|\\frac{z}{\\lambda}-v\\\|<r\\}$
	$\\displaystyle=$	$\\displaystyle\\{z\\in V:\\\|z-\\lambda v\\\|<\|\\lambda\|r\\}$
	$\\displaystyle=$	$\\displaystyle B_{\|\\lambda\|r}(\\lambda v).$