In Tate's thesis, he claims that for a local field $k$ and any non-trivial character $\chi$ of its addition group $k^+$, the correspondence $\eta\mapsto\chi(\eta\xi)$ is an isomorphism, both topological and algebraic, between $k^+$ and its character group.
He uses 6 steps to prove this, and what is confusing me is in the fourth, where he shows the image of the correspondence is everywhere dense in the character group. He says $\chi(\eta\xi)=1\text{ for any }\eta\Rightarrow\xi=0$, but I cannot find the relation with this fact and the density of the image.