2.4.E: Problems on Upper and Lower Bounds (Exercises)

Exercise

Complete the proofs of Theorem 2 and Corollaries 1 and 2 for infima.
Prove the last clause of Note

Exercise

Prove that is complete iff each nonvoid left-bounded set in has an infimum.

Exercise

Prove that if are right bounded (left bounded) in so is

Exercise

Prove that if is an open interval then

Exercise

In an ordered field let Let and let denote the set of all products i.e.,


In both cases, assume that the right-side sup (respectively, inf exists.

Exercise

From Problem 5 with obtain a new proof of Theorem 1.
[Hint: If is left bounded, show that is right bounded and use its supremum.

Exercise

Let and be subsets of an ordered field Assuming that the required lub and glb exist in prove that
(i) if then ;
(ii) if then ;
(iii) if then .

Exercise

For any two subsets and of an ordered field let denote the set of all sums with and i.e.,

Prove that if and exist in then

similarly for infima.
[Hint for sup: By Theorem we must show that
(i) and
(ii') .
Fix any By Theorem 2,

Then

as required.

Exercise

In Problem 8 let and consist of positive elements only, and let

Prove that if and exist in then

similarly for infima.
[Hint: Use again Theorem 2 For take

and

show that

For inf let and choose with

Now take and with

and show that

Explain!

Exercise

Prove that
(i) if then ;
(ii) if then .

Exercise

Prove the principle of nested intervals: If are closed intervals in a complete ordered field with

then

[Hint: Let

Show that is bounded above by each .
Let (Does it exist?)
Show that

i.e.,

Exercise

Prove that each bounded set in a complete field is contained in a smallest closed interval (so is contained in any other .
Show that this fails if "closed" is replaced by "open."
.

Exercise

Prove that if consists of positive elements only, then iff
(i) and
(ii) .
[Hint: Use Theorem 2.