Let $| \ \ |_ 1$ and $| \ \ |_ 2$ be absolute values on $k$. Prove that the topologies on $k$ induced by $| \ \ |_ 1$ and $| \ \ |_ 2$ coincide if and only if $| \ \ |_ 1$ and $| \ \ |_ 2$ are equivalent.
Here $k$ is a field and equivalent norm means sequences that are cauchy in one are cauchy in another and similarly convergent sequences in one are convergent in another. I know I am supposed to use Weak approximation lemma. (This is supposed to be a corollary of the weak approximation lemma)
I think $(\implies)$ is easy.
For the other way, I do not understand how weak approximation lemma should make its appearance.
Weak approximation lemma:
Given inequivalent nontrivial absolute values $| \ |_ 1 , . . . , | \ |_ n$ on $k$, $a_ 1 , . . . , a_ n ∈ k$, and $ \epsilon _1 , . . . , \epsilon_ n ∈ \mathbb R_{ >0}$ there exists $x ∈ k$ such that $|x − a_ i | _ i < \epsilon _i$ for $i = 1, . . . , n.$