Transitive Closure of Reflexive Relation is Reflexive

Theorem
Let $S$ be a set.

Let $\mathcal R$ be a reflexive relation on $S$.

Let $\mathcal T$ be the transitive closure of $\mathcal R$.

The $\mathcal T$ is reflexive.

Proof
Let $a \in S$.

Since $\mathcal R$ is reflexive:
 * $\left({a,a}\right) \in \mathcal R$

By the definition of transitive closure:
 * $\mathcal R \subseteq \mathcal T$

Thus by the definition of subset:
 * $\left({a,a}\right) \in \mathcal T$

Since this holds for all $a \in S, $\mathcal T$ is reflexive.