I am interested in the generalization of the well-known theorem of norm equivalences for finite dimensional spaces over $\mathbb R$ or $\mathbb C$. However, I have trouble understanding the proof I found in a lecture notes.
We fix $K$ a valued field whose norm is denoted by $|\cdot|$. Let $V$ be a finite dimensional $K$-vector space. We shall prove that all norms over $V$ are equivalent. For this, we use induction on $d=\dim(V)$. We fix a basis $e_1,\ldots,e_d$ of $V$ and we consider the sup norm $||\cdot||_{\infty}$ with respect to it. We also fix a norm $||\cdot||$ on $V$. It is enough to prove that $||\cdot||$ is equivalent to $||\cdot||_{\infty}$.
The result is clear when $d=1$, given that $||\cdot||$ is "$K$-linear" with respect to the valuation $|\cdot|$ over $K$.
For the induction step, at first it is clear that $||\cdot|| \leq C||\cdot||_{\infty}$ where $C=\sup(||e_i||)$. As for the reverse inequality, assume towards a contradiction that there does not exist any $D>0$ such that $||\cdot||_{\infty} \leq D||\cdot||$. The lecture notes I use now state the following:
There exists a sequence of vectors $u_n$ such that $||u_n||_{\infty} \geq 1$ for every $n$ and $||u_n||\rightarrow 0$.
I can't see how to build such a sequence $(u_n)$.
Considering the statement with $D=n$, I can build a sequence of vectors $v_n\in V$ such that $$||v_n||_{\infty} > n||v_n||$$
Now, if the base field were $\mathbb R$ or $\mathbb C$ as in the usual theorem, I could define $u_n := \frac{v_n}{||v_n||_{\infty}}$ and it would work. But because the base field may not contain the element $||v_n||_{\infty}\in \mathbb R$, this reasoning won't do here.
Would you have an idea as to how to construct a suitable sequence $(u_n)$ ?
NB: once such a sequence is constructed, it is not so difficult to conclude the proof using the induction hypothesis. I omit it here.