User:Abcxyz/Sandbox/Real Numbers/Real Addition is Associative

Theorem
Let $\R$ denote the set of real numbers.

Let $+$ denote addition on $\R$.

Then $+$ is associative on $\R$.

Proof 1
Let $\left({\R, +, \times, \le}\right)$ denote the real numbers, as axiomatically defined as a Dedekind complete totally ordered field.

By the field axioms, $+$ is associative on $\R$.

Proof 2
Let $\R$ denote the set of real numbers, as constructed from Cauchy sequences.

Let $+$ denote addition on $\R$.

From Rational Addition is Associative, it directly follows that $+$ is associative on $\R$.

Proof 3
Let $\R$ denote the set of real numbers, as constructed from Dedekind cuts.

Let $+$ denote addition on $\R$.

From Rational Addition is Associative, it directly follows that $+$ is associative on $\R$.

Proof 4
Let $\R$ denote the set of real numbers, as defined as the Dedekind completion of the rational numbers.

Let $+$ denote addition on $\R$.

By definition, $\left({\R, +}\right)$ is a group.

By the group axioms, $+$ is associative on $\R$.