Symmetric Transitive and Serial Relation is Reflexive

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

$$\forall x \in S: \exists y \in S: x \mathcal{R} y$$

Then $$\mathcal{R}$$ is reflexive.

Proof
As $$\mathcal{R}$$ is symmetric, $$x \mathcal{R} y \Longrightarrow y \mathcal{R} x$$.

As $$\mathcal{R}$$ is transitive, $$x \mathcal{R} y \land y \mathcal{R} x \Longrightarrow x \mathcal{R} x$$.

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.