My problem is that my method seems long-winded, so I wanted to know if I was overlooking something obvious.
Background: Definition: A valuation is a real-valued function $|.|$ on a field $k$ such that (1)$|b|\geq 0$ with equality only for $b=0$. (2)$|bc|=|b||c|$ for all $b,c\in k$. (3)There is some $C \in \mathbb{R}$ such that $|b|\leq 1 \Rightarrow |1+b| \leq C$.
Problem: On page 22 of Local Fields, Cassels writes that for $b\in k$ with $|b|_1<1$, "we have $\frac{b^n}{1+b^n}\rightarrow 0$".
To show this, I considered a valuation $|.|$ equivalent to $|.|_1$ that satisfies the triangle inequality (every valuation is equivalent to one satisfying the triangle inequality). Its easy to show $\frac{b^n}{1+b^n}\rightarrow 0$ with respect to this $|.|$ valuation. Then since equivalent valuations induce the same topology, this means $\frac{b^n}{1+b^n}\rightarrow 0$ with respect to $|.|_1$. Is this making a mountain out of a mole hill?