Reflexive Relation is Serial

Theorem
Every reflexive relation is also a serial relation.

Proof
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

We have that $\mathcal R$ is serial :
 * $\forall x \in S: \exists y \in S: \left({x, y}\right) \in \mathcal R$

That is, every element relates to at least one element.

We have that $\mathcal R$ is reflexive :
 * $\forall x \in S: \left({x, x}\right) \in \mathcal R$

Hence if $\mathcal R$ is reflexive, every $x$ is related to itself, thereby fulfilling the criterion for $\mathcal R$ to be serial.