I don't understand how to prove a conclusion in the Theorem.
When k is $p$-adic, the subgroups 1+$p^{v}$, $v>0$, of $u$ $(|u|=1)$ form a fundamental system of neighborhoods of $1$ in $u$, We must have therefore $\tilde{c}(1+p^{v})$ = 1 for sufficiently large $v$, $\tilde{c}$ is a character of $u$.
Why do the characters of $u$ equal to 1 in a neighborhood of $1$ in $u$?
Is it related to topology of $u$ ?
Thank you in advance.
