So I want to proove a theorem I read in Vladimir Berkovich's "Spectral Theory and analytic geometry over non-achimedian fields".
The theorem is : Let M be an abelian group, N a subgroup of M, and $\vert . \vert$ a semi-norm on M. The residue semi-norm is a norm if and only if N is closed
And a semi-norm $\vert.\vert : M \rightarrow \mathbb{R}_{+}$ is so that :
$\forall (a,b) \in M^2, \vert a-b \vert \leq \vert a \vert +\vert b \vert$
$\vert 0_M \vert = 0$
A norm is a semi-norm such that : $\vert a \vert=0 \implies a=0_M$
The residual semi-norm $\vert . \vert_{G/N}$ is such that : $\vert\pi(a)\vert_{G/N}=$inf$ ( \vert u \vert, \pi(u)=\pi(a) )$
I managed to proove that the residual semi-norm is a semi-norm. I am now stuck on the theorem.
I began trying to proove that : If N is closed and $g \notin \pi^{-1}(N)$ then, $\vert \pi(g) \vert \neq 0$. Appart from writing that $g$ is in $^cN$, which is open, so there is a ball of which $g$ is the center in $^cN$, I haven't been able to do much : I haven't found a thing to say about that ball.
I also tried ad absurdo but it didn't yeald.
Sorry for mistakes, english is not my first language. Thanks for any help!