[Few notes] clear definition of Dedekind cutting theorem

1 , theorem content

Dedekind cutting theorem: Set is a cut of the real number set, or there is a maximum number, or there is a minimum number.

Definite definition: The number set of the upper bound must have an upper bound, and the number set of the lower bound must have a definite boundary.

2. Certification process

Set of non-null sets has upper bound

Remember, that is, the set of upper bounds

The complement of the order is the set of

Thus forming a cut of the real number set

Known by the Dedekind theorem, there is either a maximum number or a minimum number

If there is a maximum number, set the maximum number

Because, so not the upper bound

Thus,s.t

Well, thus is not the upper bound, so

With the maximum number of contradictions, thus no maximum number

So there's a minimum number

There is a minimum upper bound, that is, upper bound #

[Few notes] clear definition of Dedekind cutting theorem