I read in the book Applied Analysis by Hunter and Nachtergale that
the sequence $x(n)=\log(n)$ is not Cauchy since $\log(n)\to\infty$
But that seems to be irrelevant to the definition of a Cauchy sequence which I understand is as follows:
A sequence $x(n)$ is said to be Cauchy if for every $\epsilon > 0$ there exists an $N$ such that $\lvert x(m)-x(n)\rvert < \epsilon$ for all $m,n>N$.
This sequence $(\log n)$ seems to meet the definition. So how come it is not considered Cauchy?