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, $$x \mathcal R y \and y \mathcal R x \implies x \mathcal R x$$.

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.