Finding a sequence a with $\lim_{ n\to ∞} (a_{n+1}-a_n)=0$ a:divergent

Question

Question is in the title. I would appreciate any help with this as I am a bit clueless.

Answer 1

Note that if $a_{n}=H_{n}$ where $H_{n}$ denotes the $n$-th harmonic number, $$\lim_{n \to \infty} H_{n+1}-H_{n}=\lim_{n \to \infty} \frac{1}{n+1}=0$$

However, the Harmonic Series is divergent.

Answer 2

Take $a_n=\ln n$. Then $$\lim_{n\to\infty}\Big[\ln(n+1)-\ln(n)\Big]=\lim_{n\to\infty}\ln\frac{n+1}{n}=\ln\left[\lim_{n\to\infty}\frac{n+1}{n}\right]=\ln 1=0$$