Constructing the real numbers Intro real analysis Exhibit t

Exhibit two nonempty subsets A and B of Q surti that a b for every a A and b B, but there exists no r Q such that a r b for every a A and b B. show that there exists a real number r such that a r b for ever a A and b B. Show that if, in addition to the assumptions in part (b), we also have A B = R, then there exists a unique number r as in part (b).

Solution

a) A and B are two subsets of Q.

Given that a<=b for all a in A and b in B.

This implies that set A is bounded above and set B is bounded below

There is a supremum l for A and m infremum for B.

If supremum of A = infremum of B, and they do not belong to A and B

then highest element in A<inf and sup < lowest element in B.

Thus proved

They are equal if both belong to the sets

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b) Same proof holds good for real numbers also.

c) If AUB =R

then it implies that A and B are mutually exhaustive.

There is no element in r which is not in A and B.

Hence supremum of A being a real number must be either in A or in B.

If it is in A, then that is infremum for B, and this r is unique.

Similarly if it is in B, then that is supremum of A.