Addition of Cuts is Associative

Theorem

Let $\alpha$, $\beta$ and $\gamma$ be cuts.

Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$.

Then:

$\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$

Proof

$\paren {\alpha + \beta} + \gamma$ is the set of all rational numbers of the form $\paren {p + q} + r$ such that $p \in \alpha$, $q \in \beta$ and $r \in \gamma$.

Similarly, $\alpha + \paren {\beta + \gamma}$ is the set of all rational numbers of the form $p + \paren {q + r}$ such that $p \in \alpha$, $q \in \beta$ and $r \in \gamma$.

From Rational Addition is Associative we have that:

$\paren {p + q} + r = p + \paren {q + r}$

The result follows.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.14$. Theorem $\text {(b)}$