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I've always intuitively understood this set in intuitive sense, as "all numbers on the number line". However, now I want to know the formal definition:

Consider the set of rational numbers, $\mathbb{Q}$.

For any two Cauchy sequences of rational numbers $X=⟨x_n⟩,Y=⟨y_n⟩$, define an equivalence relation between the two as:

$X≡Y⟺∀ϵ>0:∃n∈ \mathbb{N}:∀i,j>n:|x_i−y_j|<ϵ$

The real numbers are the set of all equivalence classes $[[⟨x_n⟩]]$ of Cauchy sequences of rational numbers.

Most importantly, how do we deduce the axioms of the real number field from this definition?

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    $\begingroup$ What, more concretely, are the axioms you want to confirm that this $\Bbb R$ fulfills? $\endgroup$ – Arthur Jan 22 '16 at 15:53
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    $\begingroup$ As a side note, this is a construction of $(\mathbb{R},|\cdot|)$ by a completion of $(\mathbb{Q},|\cdot|)$. There is an alternative construction via the method of Dedekind cuts that might prove more intuitive, as it relies more on order. $\endgroup$ – parsiad Jan 22 '16 at 15:58
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    $\begingroup$ @par It is not necessarily more intuitive. For instance, the definition of multiplication on $\Bbb R$ under the Dedekind cut construction is quite complicated compared to multiplication under the Cauchy sequence construction. $\endgroup$ – Arthur Jan 22 '16 at 16:03
  • $\begingroup$ @Arthur and it's not necessarily less intuitive either:) For instance, the existence of the least upper bound of a bounded set is easier to prove using Dedekind cuts. It's like squeezing a bag of water: you can displace the complexity, but in each case you have to sweat some details. $\endgroup$ – BrianO Jan 22 '16 at 17:03
  • $\begingroup$ I meant that it might prove to be more intuitive [to the author] :-) $\endgroup$ – parsiad Jan 22 '16 at 17:37

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