Transitive, Symmetric, Non-Reflexive Relation

Can anyone give me an example of a symmetric, transitive but non-reflexive relation with the set S as any integer? I can only think of the sibling example, which doesn't work since I need integers, and the empty relation, which is rather cheap...Recipricol too, but it needs real numbers and not integers. Blllarrrrgh!

In addition, why is this proof not valid?

Suppose R is a symmetric and transitive relation. aRb means bRa by the symmetric property. By the transitive property, aRb and bRa means aRa, so the relation must also be reflexive.

Since the sibling example exists, I know for sure it's wrong. But I can't see what it doesn't take into account. Perhaps the chance that the property of the relation forces the transitive property to use three different numbers?

Thanks in advance.