I'm trying to understand a lemma from the beginning of John Tate's PhD thesis. Here $k$ is a local field of characteristic zero. The character group $\hat{k}$ of $k^+$ is the group of continuous homomorphisms $k^+ \rightarrow S^1$, with the compact open topology, which has subbasis
$$V(K,U) = \{ \chi \in \hat{k} : \chi(K) \subseteq U \}$$
for $K$ compact and $U$ open. Since $S^1$ is metrizable and $k$ is a union of increasing compact subsets, each contained in the interior of the next, the group $\hat{k}$ is metrizable, and $f_n \to f$ in this metric if and only if $f_n \to f$ uniformly on compact sets.
For property (4), it seems all that has been proven is that the map $k \rightarrow \hat{k}$ is injective. What does this have to do with the image of $k$ being dense?