Real Multiplication is Associative

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Theorem

The operation of multiplication on the set of real numbers $\R$ is associative:

$\forall x, y, z \in \R: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$


Proof

From the definition, the real numbers are the set of all equivalence classes $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.


Let $x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right], z = \left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]$, where $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$, $\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$ and $\left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]$ are such equivalence classes.

From the definition of real multiplication, $x \times y$ is defined as $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right]$.


Thus we have:

\(\displaystyle x \times \left({y \times z}\right)\) \(=\) \(\displaystyle \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left({\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n \times z_n} \right \rangle}\right]\!\right]\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{\left \langle {x_n \times \left({y_n \times z_n}\right)} \right \rangle}\right]\!\right]\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{\left \langle {\left({x_n \times y_n}\right) \times z_n} \right \rangle}\right]\!\right]\) $\quad$ Rational Multiplication is Associative $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{\left \langle {x_n \times y_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] \times \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]}\right) \times \left[\!\left[{\left \langle {z_n} \right \rangle}\right]\!\right]\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x \times y}\right) \times z\) $\quad$ $\quad$

$\blacksquare$


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