Congruence Modulo Real Number is Equivalence Relation

Theorem
For all $z \in \R$, congruence modulo $z$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexive
We have that Equal Numbers are Congruent:
 * $\forall x, y, z \in \R: x = y \implies x \equiv y \pmod z$

so it follows that:
 * $\forall x \in \R: x \equiv x \pmod z$

and so congruence modulo $z$ is reflexive.

Symmetric
So congruence modulo $z$ is symmetric.

Transitive
So congruence modulo $z$ is transitive.

So we are justified in supposing that congruence, as we have defined it, is an equivalence.

Also see

 * Congruence Modulo Integer is Equivalence Relation, in which context this result is usually encountered