Let us consider a map $\beta:V\setminus\{0\}\to{\mathbb{R}}_{>0}/|k^{\times}|$ given by

$$\beta(v)=\text{the class of $\|v\|$ in ${\mathbb{R}}_{>0}/|k^{\times}|$}.$$

It is sufficient to see that $\beta(V\setminus\{0\})$ is finite. Let $\beta_{1},\ldots,\beta_{l}$ be distinct elements of $\beta(V\setminus\{0\})$. We choose $v_{1},\ldots,v_{l}\in V\setminus\{0\}$ with $\beta(v_{i})=\beta_{i}$ for $i=1,\ldots,l$. If $i\not=j$, then $\|a_{i}v_{i}\|\not=\|a_{j}v_{j}\|$ for all $a_{i},a_{j}\in k^{\times}$. Therefore, we obtain

$$\|a_{1}v_{1}+\cdots+a_{l}v_{l}\|=\max\{\|a_{1}v_{1}\|,\ldots,\|a_{1}v_{l}\|\}$$

for all $a_{1},\ldots,a_{l}\in k$. In particular, $v_{1},\ldots,v_{l}$ are linearly independent. Therefore, we have $\#(\beta(V\setminus\{0\}))\leq\dim_{k}V$. ∎