Q:

Transitive, symmetric $R$ such that for all $x$, there is $y$ such that $xRy$, is an equivalence relation

I'm stuck at one particular task I'm working on.
Here is the task:

Let R be a transitive and symmetrical relation on $S$. Assume that for all 
  $x \in S$ there is a $y \in S$ so that $xRy$. Prove that $R$ is an equivalence 
  relation.

How can I prove that $R$ is an equivalence relation?
I would appreciate any help.
Thank you.

A:

All you need to do here is make the case that $R$ must be reflexive. Since for every $a \in S$ there is $b,$ such that $aRb,$ then by symmetry $bRa$ and by transitivity $aRa.$ Since this holds for every $a\in S$, the relation is necessarily reflexive, and hence, is an equivalence relation.

