Q: Prove this is not a Cauchy sequence If $$ x_{n}:= \sqrt{n}$$ show that$$ x_{n}$$ satisfies $$ \lim_{n\to \infty}|x_{n+1}-x_{n}|=0 $$ but that it is not a Cauchy sequence. 

A: $$|x_{n+1}-x_n|=\sqrt{n+1}-\sqrt n=\frac1{\sqrt{n+1}+\sqrt n}\xrightarrow[n\to\infty]{}0$$
But since $\;\sqrt n\xrightarrow[n\to\infty]{}\infty\;$  the sequence doesn't converge finitely, which is a necessary and sufficient condition for a sequence to be Cauchy..
