Show that the set of all real numbers is an abelian group with respect to addition?

Let R represent the set of real numbers. Then

Closure

For all x and y in R, x+y belongs to R.

Associativity

For all x, y and z in R, (x + y) + z = x + (y + z).

Identity element

There exists an element in R, denoted by 0, such that for every x in R, x + 0 = x = 0 + x.

Inverse element

For each x in R, there exists an element y in Rsuch that x + y = 0 = y + x where 0 is the identity element (defined above). y is denoted by -x.

The above proves that R is a group.

Commutativity

For any x and y in R, x + y = y + x.

The group is, therefore, Abelian.