A sequence that satisfies the condition: $\forall p \in \mathbb{N}: \lim_\limits{n \to \infty } | a_p - a_{p+1}| = 0$ but is not a cauchy sequence.

Question

I need to find a sequence that satisfies the condition: $\forall p \in \mathbb{N}: \lim_\limits{n \to \infty } | a_n - a_{n+p}| = 0$ but is not a cauchy sequence.

This somehow implies for me that I need a convergent sequence to satisfy the condition, however this contradicts the second condition of the task, namely that an is not a Cauchy sequence.

So there clearly is something else I'm not taking into account but I can't figure out what. Please help me.

Answer 1

Define $$a_n =\sum_{k=1}^n \frac{1}{k}$$ Clearly, $\{a_n\}$ isn't convergent but it satisfies the condition you want.