Let $ R $ be a relation on a set $ A $, and suppose that $ R $ is symmetric and transitive. Show that if for every $ x \in A $, there exists $ y \in A $ such that $ xRy $, then $ R $ is an equivalence relation.
My attempt
What you really have to prove is that $ R $ is reflexive, that is $ xRx $. So, let $ x \in A $, then there exists $ y \in A $ such that $ xRy $.
Since $ R $ is symmetric, we have $ yRx $.
In this way we have that for $ x \in A $, there exists $ y \in B $, such that $ xRy $ y $ yRx $, by transitivity we have $ xRx $, that shows that it is reflexive, therefore a relation of equivalence.