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|.

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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?