Real Analysis – Pseudo-Cauchy Sequence

Question

I have never seen this terminology before, so I will provide the given definition.

A Pseudo-Cauchy sequence is : A sequence $(a_n)$ if for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that $|a_{n+1} – a_n | \leq \epsilon \space \forall \space n \geq N$

So then my question is that is a pseudo-cauchy sequence always converging?

Answer 1

Take $$a_n = \sum_{k=1}^n {1\over k}.$$