Transitive Closure of Reflexive Relation is Reflexive

Theorem
Let $S$ be a set.

Let $\RR$ be a reflexive relation on $S$.

Let $\RR^+$ be the transitive closure of $\RR$.

Then $\RR^+$ is reflexive.

Proof
Let $a \in S$.

Since $\RR$ is reflexive:
 * $\tuple {a, a} \in \RR$

By the definition of transitive closure:
 * $\RR \subseteq \RR^+$

Thus by the definition of subset:
 * $\tuple {a, a} \in \RR^+$

Since this holds for all $a \in S$, it follows that $\RR^+$ is reflexive.

Also see

 * Reflexive Closure of Transitive Relation is Transitive