TITLE: Pseudo-Cauchy sequence
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: Sequence $$a_n = \sqrt n$$ is pseudo-Cauchy but divergent. It is monotonic, strictly increasing. but not bounded, and divergent.
Let it through the sine function and we get another pseudo-Cauchy sequence $$b_n = \sin(\sqrt n),$$ which is bounded (oscillating between $-1$ and $1$) but not convergent, either.