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I am teaching my self real analysis from Pugh's Real Mathematical Analysis, and I came acrosse the construction of reals through Dedekind cuts, but am confused about the following:

Dedekind cuts are defined as follows ( Pugh's page 11):

A cut in ##Q## is a pair of subsets ##A,B ## of ##\mathbb{Q}~## such that:

1) ##A\cup B = \mathbb {Q},~A\not=\phi,~B\not=\phi,~A\cap B=\phi.##

2) if ##a\in A## and ##b\in B## then ##a<b##.

3) ##A## contains no largest element.

Then few lines below he make the following definition:

A real number is a cut in ##\mathbb{Q}##.

What I don't understand is how a pair of sets ( which are infinite ) constitutes a real number.

I would like somebody to give me, if possible, a brief explanation about Dedekind cuts, too.

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# I Confused about Dedekind cuts

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