26 1. The Real Numbers

is a one-to-one function from N onto this subset. By definition it takes the

successor k + 1 of an element k ∈ N to the successor nk + 1 of its image nk.

6. Let F be an ordered field. We consider N to be a subset of F as described in

the preceding exercise. Prove that F is Archimedean if and only if, for each

pair x, y ∈ F with x 0, there exists a natural number n such that nx y.

7. Prove that if x y are two real numbers, then there is a rational number r

with x r y. Hint: Use the result of Example 1.4.9.

8. Prove that if x is irrational and r is a non-zero rational number, then x+r and

rx are also irrational.

9. We know that

√

2 is irrational. Use this fact and the previous exercise to prove

that if r s are rational numbers, then there is an irrational number x with

r x s.

The following exercises concern Dedekind cuts of the rationals and should be done

using only properties of the rational number system and the definition of Dedekind

cut.

10. Show that if Lx and Ly are Dedekind cuts defining real numbers x and y, then

Lx + Ly = {r + s : r ∈ Lx and s ∈ Ly}

is also a Dedekind cut (this is the Dedekind cut determining the sum x + y).

11. If Lx and Ly are Dedekind cuts determining positive real numbers x and y and

if we set

K = {rs : 0 ≤ r ∈ Lx and 0 ≤ s ∈ Ly} ∪ {t ∈ Q : t 0},

then K is also a Dedekind cut (this is the Dedekind cut determining the product

xy).

12. If L is the Dedekind cut of Example 1.4.2 and L determines the real number x

(so that L = Lx), prove that Lx2 = L2. Thus, the real number corresponding

to L has square 2.

1.5. Sup and Inf

The concept of least upper bound, which appears in the completeness axiom, will

be extremely important in this course. It will be examined in detail in this section.

We first note that there is a companion concept for sets that are bounded below.

Greatest Lower Bound. We say a set A is bounded below if there is a number

m such that m ≤ x for every x ∈ A. The number m is called a lower bound for A.

A greatest lower bound for A is a lower bound that is larger than any other lower

bound.

Theorem 1.5.1. Every non-empty subset of R that is bounded below has a greatest

lower bound.