Need an example where the relation R isn't transitive

Question

R is a relation on set A. $$S=\bigcup_{n=1}^{n}R^n$$ I want an example where works for every n∈N (N natural numbers) that relation S is not a transitive.

Answer 1

Let $R=\{\,(k+1,k)\mid k\in \Bbb Z\,\}$. Then $S$ is the relation on $\Bbb Z$ describable as "is greater than, but at most by $n$". Clearly, $2n \mathrel S n$ and $n \mathrel S 0$, but $2n\not\mathrel S0$.