A metric space $(X,d)$ is called ultrametric if strict triangle inequality holds: $$d(x,y)\le \max\{ d(x,z), d(z,y)\}.$$
In the book $p$-adic Lie groups by Schneider, the author states following theorem (p.6):
A sequence $(x_n)$ in $X$ is a Cauchy sequence if and only if $\lim_{n\rightarrow \infty} d(x_n,x_{n+1})=0$.
I confused here whether this is true only for ultrametric space $(X,d)$? With usual metric on $\mathbb{R}$, we can take $x_n=1+\frac{1}{2}+\cdots + \frac{1}{n}$, which is not a Cauchy. Am I missing something?
There are questions on mathstack where the condition in above statement is replaced by $d(x_n,x_{n+1})<Cr^n$ for some fixed $C\in\mathbb{R}$ and $0<r<1$ for all $n\ge 1$. Then $(x_n)$ is Cauchy.
It seems that this condition in terms of $C$ and $r$ is weaker than that in above quoted statement. Am I right? In other words, we have only one way implication below but not conversely: $$ \Big{(}d(x_{n+1},x_n)<Cr^n \mbox{ for some $r$, $0<r<1$ and fixed $C\in\mathbb{R}$}\Big{)}\Rightarrow \lim_{n\rightarrow\infty}d(x_{n+1},x_n)=0. $$
