Real Addition is Associative

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Theorem

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

$\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$


Proof

From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.


Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\sequence {y_n} } {}$ and $\eqclass {\sequence {z_n} } {}$ are such equivalence classes.

From the definition of real addition, $x + y$ is defined as $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$.

Thus we have:

\(\displaystyle x + \paren {y + z}\) \(=\) \(\displaystyle \eqclass {\sequence {x_n} } {} + \paren {\eqclass {\sequence {y_n} } {} + \eqclass {\sequence {z_n} } {} }\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n + z_n} } {}\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {\sequence {x_n + \paren {y_n + z_n} } } {}\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {\sequence {\paren {x_n + y_n} + z_n} } {}\) Rational Addition is Associative
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {\sequence {x_n + y_n} } {} + \eqclass {\sequence {z_n} } {}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} } + \eqclass {\sequence {z_n} } {}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {x + y} + z\)

$\blacksquare$


Sources