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: You'll want to show that the sequence is Cauchy. Use the fact that every number can be written as $3k$, $3k-1$, or $3k+1$ for some $k$, and use a triangle inequality argument with each term eventually bounded above by $\epsilon/3$.
For a counterexample satisfying the first condition only, consider $a_n=\log n$.
For a counterexample satisfying the second condition only, $a_n=\begin{cases}1 & \text{if }n\text{ is an integer multiple of }3\\0 & \text{otherwise.}\end{cases}$
