$R$ is a relation on set $U$, Given that $R$ is transitive.
Need to prove $R^{n+1} \subseteq R^n$ for all $n \geq 1$
I tried it by Induction
base case for $n=1$
$ R^{2}$ $= R \circ R $ $\qquad$ { Def of exponents }
$\subseteq R $ $\qquad$ { If $R$ is transitive $\Rightarrow (R \circ R) \subseteq R $ }
Now took $I.H$ step as $R^{n+1} \subseteq R^n$
Took $n=n+1$
$R^{n+2} = R \circ R \circ R^n$ $\qquad$ { Def of exponents }
$\subseteq R \circ R^n $ $\qquad \qquad \qquad $ { as If $R$ is transitive $\Rightarrow (R \circ R) \subseteq R $, and using monotonicity }
$=R^{n+1}$
Now my question is that I havn't used the $I.H$ step here and was able to prove it just by using $R$ is transitive $\Rightarrow (R \circ R) \subseteq R $
But below is the prove If I would have done it by using $I.H$
$R^{n+2} = R \circ R^{n+1}$ $\qquad$ { Def of exponents }
$\subseteq R \circ R^n $ $\qquad \qquad \qquad $ { $I.H$ step + monotonicity }
$=R^{n+1}$
So which way is correct ?