Why is the sequence $x(n) = \log n$ **not** Cauchy? 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 $\epsilon &gt; 0$  there exists an $N$ such that $\lvert x(m)-x(n)\rvert &lt; \epsilon$ for all $m,n&gt;N$.


This sequence $(\log n)$ seems to meet the definition. So how come it is not considered Cauchy?
Hint. Look at the difference
$$
\log(2n) - \log(n) .
$$
Because it can not converge. "$x_n$ is a Cauchy sequence if and only if it is convergennt".
($x_n$ is a real sequence).