I was reading the section on Completions in Neukirch's Algebraic Number Theory.
Neukirch uses the term multiplicative valuation for what other authors seem to call absolute value. He uses the term exponential valuation for what other authors call valuation. Anyway, I will stick to his conventions.
On page 126 Neukirch mentions the following:
Let $v$ be an exponential valuation of a field $K$. Then $v$ is canonically continued to an exponential valutation $v^{\wedge}$ of the completion $K^{\wedge}$, by setting $v^{\wedge}(a) = \lim_{n \to \infty} v(a_n)$, where $a = \lim_{n \to \infty} a_n \in K^{\wedge}$, $a_n \in K$.
Perhaps this is obvious, but why does $\lim_{n \to \infty} v(a_n)$ exist?