Does $b_n = a_{n+1} - a_n $ is a zero sequence implies $a_n$ is convergent? 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.
$$a_n = \sum_{k=1}^n \frac 1 k$$
Another nice example.  $a_n = \sqrt{n}$.  Show that
$$
\sqrt{n+1} - \sqrt{n} \to 0\qquad\text{but}\qquad
\sqrt{n} \to +\infty
$$