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$\mathbb{R}$ is generally constructed as equivalence classes of Cauchy sequences. As Cauchy Completeness and Archimedean Property together are equivalent to the Bolzano-Weierstrass Theorem, there should be a method to construct $\mathbb{R}$ from $\mathbb{Q}$ using the Bolzano-Weierstrass Theorem.

So, is there such a method? If so, could someone point me towards some resources related to this?

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    $\begingroup$ A corresponding Mathematics post: Constructing Real Numbers using Bolzano-Weierstrass Theorem? $\endgroup$
    – Martin Sleziak
    Commented May 8, 2020 at 11:20
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    $\begingroup$ That's my post. I didn't get a response there, so I've asked here. $\endgroup$
    – Ishan Deo
    Commented May 8, 2020 at 11:47
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    $\begingroup$ Yes, it is recommended to link the posts to each other when cross-posting. That's why I commented under both psots. $\endgroup$
    – Martin Sleziak
    Commented May 8, 2020 at 12:06
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    $\begingroup$ I think the difficulty is that, whereas a Cauchy sequence or a Dedekind cut determines a single real, a bounded sequence determines, by Bolzano-Weierstrass, a nonempty set of reals, namely the set of accumulation points. So to produce a definition of the reals, you'd need to pick out one real from that set. It can certainly be done; e.g., the set has a largest element. But it looks ugly to me. $\endgroup$
    – Andreas Blass
    Commented May 8, 2020 at 13:03
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    $\begingroup$ "Didn't get an answer" in 13 hours? I recommend allowing 3 days before concluding you won't get an answer. $\endgroup$
    – Gerald Edgar
    Commented May 8, 2020 at 13:33

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