## Sequence in a non-Archimedean field

Let K be a field with non-Archimedean valuation ||.

Suppose that $s_n \to s$ in K, ||. Show that $|s_n| \to |s|$ in $\mathbb{R}, |\,|_\infty$.

When $s \neq 0$, show that there exists N such that for all n > N, $|s_n| = |s|$.

____________

It's the very last part I'm having trouble with. How is this supposed to follow if || isn't a discrete valuation?