Cantor's construction of the real numbers/Definition

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Definition

The set of real numbers, $\mathbb{R}$, is the quotient space, $\mathscr{C}/\sim$ where:

• $\mathscr{C}$ - the set of all Cauchy sequences in $\mathbb{Q}$ - the quotients
• $\sim$ - the usual equivalence of Cauchy sequences

We further claim:

1. that the familiar operations of addition, multiplication and division are well defined and
2. by associating $x\in\mathcal{Q}$ with the sequence $({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q}$ where $\forall n\in\mathbb{N}[x_n:=x]$ we can embed $\mathbb{Q}$ in $\mathbb{R}:=\mathscr{C}/\sim$