Prove the convergence My task is to prove, that if this both conditions hold$$\lim_{n\rightarrow\infty}(a_{n+1}-a_n)=0$$ $$\forall{\varepsilon >0} \exists{N\in\mathbb{N}}\forall{n,m>N}: |a_{3m}-a_{3n}|\leq\varepsilon$$then $a_{n}$ converges, and show examples, that none of above is alone sufficient for $a_{n}$ to converge.
I am afraid I don't understand second condition good enough to do anything. It seems similiar to Cauchy's condition of limit, but if it is so, then $a_{3n}$ is a kind of limit for $a_{3m}$. But what does it mean that subseries are bounded by themselves? What about examples? I would greatly appreciate any help and I look forward to it. Thank you  in advance

Hint: any $n \in \mathbb{N}$ can be written either as $3k$, or $3k+1$, or $3k+2$ for some $k \in \mathbb{N}$. The sequence $\{a_{3k}\}_k$ converges. Moreover
$$
|a_{3k+1}-a_{3k}| \leq \varepsilon
$$
and
$$
|a_{3k+2}-a_{3k}| \leq |a_{3k+1+1}-a_{3k+1}| + |a_{3k+1}-a_{3k}| \leq 2 \varepsilon
$$
when $k \gg 1$.
