I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy [duplicate]
Solution 1:
Try $$ x_n =\sum_{k=1}^n \frac{1}{k}. $$
A slightly different example would be $x_n =\ln n$.
Solution 2:
Another example is $$x_n = \sin \sqrt n \, .$$ As in Prove the divergence of the sequence $\left\{ \sin(n) \right\}_{n=1}^{\infty}$., the set of limit points is the entire interval $[-1, 1]$, and using the mean value theorem it is easy to see that $$ | x_{n} - x_{n-1} | \le \frac {1}{2 \sqrt {n-1}} \to 0 \, . $$