Equivalence between different definitions of transitive closure

Question

Let $W$ be an arbitrary, non-empty set and let $R$ be an arbitrary binary relation on $W$. Define the transitive closure of $R$ as $R^+ = \bigcap \{ R' \; | \; R' \text{ is a binary transitive relation on } W \text{ and } R \subseteq R'\}$. Now consider an arbitrary pair $\langle u, v \rangle$. I want to show that $R^+uv$ iff there is a sequence of elements $u=w_0, w_1, \dots, w_n = v$ from $W$ such that for each $i < n$, $Rw_iw{i+1}$ (thus proving that the definition of transitive closure by taking the intersection is equivalent to the defintion which uses finite chains). From right to left it's obvious, but I'm having problems in thinking about how to prove the left to right direction. Any tips?

Answer 1

Call $R^+$ the relation coming from the intersection. Define $R^*$ the relation coming from the chains.

It is easy to show that $R^*$ is a transitive relation containing $R$. So $R^+ \subseteq R^*$.

Now suppose $R'$ is an arbitrary transitive relation containing $R$. Note that $R^* \subseteq R'$. This implies $R^* \subseteq R^+$.