Does $b_n = a_{n+1} - a_n $ is a zero sequence implies $a_n$ is convergent?

Question

Let $(a_n)$ be a sequence in $\mathbb{R}$ and let $b_n = a_{n+1} - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.

Answer 1

$$a_n = \sum_{k=1}^n \frac 1 k$$

Answer 2

Counterexample: $a_n=1+\frac{1}{2}+...+\frac{1}{n}$.

Answer 3

Another nice example. $a_n = \sqrt{n}$. Show that $$ \sqrt{n+1} - \sqrt{n} \to 0\qquad\text{but}\qquad \sqrt{n} \to +\infty $$