24 1. The Real Numbers

Theorem 1.4.4. If R is defined using Dedekind cuts of Q, as above, then every

non-empty subset of R which is bounded above has a least upper bound.

Proof. Let A be a bounded subset of R and let m be any upper bound for A. For

each x ∈ A, let Lx be the corresponding cut in Q. Then x ≤ m for all x ∈ A means

that Lx ⊂ Lm for all x ∈ A. We set

L =

x∈A

Lx.

Then L is a proper subset of Q because L ⊂ Lm. If r ∈ L and s r, then r ∈ Lx

for some x ∈ A and this implies s ∈ Lx and, hence, s ∈ L. If L had a largest

element t, then t would belong to Lx for some x, and it would have to be a largest

element for Lx – a contradiction. Thus, L has no largest element. We have now

proved that L satisfies (a), (b), and (c) of Definition 1.4.1 and, hence, that L is a

Dedekind cut.

If y is the real number corresponding to L, that is, if L =

Ly, then, for all

x ∈ A, Lx ⊂ Ly, and this means x ≤ y. Thus, y is an upper bound for A. Also,

Ly ⊂ Lm means that y ≤ m. Since m was an arbitrary upper bound for A, this

implies that y is the least upper bound for A. This completes the proof.

This completes our outline of the construction of the real number system begin-

ning with Peano’s axioms for the natural numbers. The final result is the following

theorem, which we will state without further proof. It will be the starting point for

our development of calculus.

Theorem 1.4.5. The real number system R is a complete ordered field.

Example 1.4.6. Find all upper bounds and the least upper bound for the following

sets:

A = (−1,2) = {x ∈ R : −1 x 2};

B = (0,3] = {x ∈ R : 0 x ≤ 3}.

Solution: The set of all upper bounds for the set A is {x ∈ R : x ≥ 2}. The

smallest element of this set (the least upper bound of A) is 2. Note that 2 is not

actually in the set A.

The set of all upper bounds for B is the set {x ∈ R : x ≥ 3}. The smallest

element of this set is 3 and so it is the least upper bound of B. Note that, in this

case, the least upper bound is an element of the set B.

If the least upper bound of a set A does belong to A, then it is called the

maximum of A. Note that a non-empty set which is bounded above always has

a least upper bound, by axiom C. However, the preceding example shows that it

need not have a maximum.

The Archimedean Property. An ordered field always contains a copy of the

natural numbers and, hence, a copy of the integers (Exercise 1.4.5). Thus, the

following definition makes sense.