How To Prove the Given Relation is Not Transitive Without Cherry Picking Examples I found this question in a problem set- Check whether the relation R in $\mathbb{R}$ defined by: $$R=\\{(a,b):a\leq b^3\\}$$ is reflexive, symmetric or transitive I found an example to show that it is not transitive- $$3<(\frac{3}{2})^3$$ $$\frac{3}{2}<(\frac{6}{5})^3$$ But: $$3>(\frac{6}{5})^3$$ So it isn't transitive But in many cases it becomes harder to find examples for instance where powers are different so is there a way to prove this in general for the given relation that it isn't transitive.

Define the relation $R_n$ with $n\ge3$ by $$aR_nb\iff a\le b^n$$ This relation is not transitive because for $a\gt2$ we have $$aR(a-1)\iff a\le(a-1)^n\\\\(a-1)R(\sqrt[n]{a-1})^n)\iff(a-1)\le(\sqrt[n]{a-1})^n=(a-1)$$ But it is clear that $a$ is not in relation with $(\sqrt[n]{a-1})^n=(a-1)$ because $a\
ot\le(a-1)$.