Let $k$ be the completion of an algebraic number field at a prime divisor $\mathfrak{p}$. We note that $k$ is locally compact. Let $k^{+}$ be the additive group of $k$ which is a locally compact commutative group.
Tate's Thesis Lemma 2.2.1 states that
If $\xi \rightarrow \chi(\xi)$ is one non-trivial character of $k^{+}$, then for each $\eta \in k^{+}$, $\xi \rightarrow \chi(\eta\xi)$ is also a character. The correspondence $\eta \leftrightarrow \chi(\eta\xi)$ is an isomorphism, both topological and algebraic, between $k^{+}$ and its character group.
The proof of this lemma is divided up into 6 steps, one step is to show that the characters $\chi(\eta\xi)$ are everywhere dense in the character group. Tate writes
$\chi(\eta\xi) = 1$, all $\eta \implies k^{+}\xi \neq k^{+} \implies \xi = 0$. Therefore the characters of the form $\chi(\eta\xi)$ are everywhere dense in the character group.
My question is: How does he get from showing that the $\xi = 0$ to the the result that the $\chi(\eta\xi)$ are everywhere dense?