Show directly that the following sequence is not cauchy :
$\displaystyle x_n=ln \ n$
I think what Shanks meant was consider that for every $\displaystyle n>1$ it is obviously true that $\displaystyle n^2>n$, but $\displaystyle \left|\ln\left(n^2\right)-\ln(n)\right|=\left|2\ln(n)-\ln(n)\right|=\left|\ln(n)\right|=\ln(n)$, which is clearly greater than any $\displaystyle \varepsilon>0$ for sufficiently large $\displaystyle n$.