Symmetric Transitive and Serial Relation is Reflexive

Theorem
Let $\mathcal R$ be a relation which is symmetric and transitive, and such that:


 * $\forall x \in S: \exists y \in S: x \mathcal R y$

Then $\mathcal R$ is reflexive.

Thus such a relation is an equivalence.

Proof
As $\mathcal R$ is symmetric, $x \mathcal R y \implies y \mathcal R x$.

As $\mathcal R$ is transitive, $x \mathcal R y \land y \mathcal R x \implies x \mathcal R x$.

But we have $\forall x: \exists y \in S: x \mathcal R y$.

Thus $\forall x: x \mathcal R x$ and therefore $\mathcal R$ is reflexive.

It follows by definition that such a relation is an equivalence relation.