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

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

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

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