Proof: $\bigcup _{n=1}^\infty R^n$ is transitive closure of $R$

Question

I have this exercise:
Let $R\subseteq A^2$ be any relation. Proove $\bigcup _{n=1}^\infty R^n$ is the transitive closure of $R$.

I have no idea what to do. Could you help me, please?

Answer 1

If $S$ is any transitive relation such that $R \subseteq S$, then show that $R^n \subseteq S$, so $\cup_{n=1}^{\infty} R^n \subseteq S$.

On the other hand, show directly that $\cup_{n=1}^{\infty} R^n$ is indeed transitive. And it certainly contains $R$ so...