A sequence $\{a_n\}$ that diverges but $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$

Question

What is a sequence $\{a_n\}$ that diverges and $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$

Answer 1

Hint: consider harmonic series.

Answer 2

Take $$ a_n=\sqrt{n} $$ then note that, for $n\ge1$, $$ a_{n+1}-a_n=\sqrt{n+1}-\sqrt{n}=\frac1{\sqrt{n+1}+\sqrt{n}}. $$

Answer 3

Besides the two good answers you already got, you could also take

$$a_n:=\log n\implies a_{n+1}-a_n=\log\left(1+\frac1n\right)\xrightarrow[n\to\infty]{}\;\ldots$$