You may find it useful to think of as points on a line, and as points in a plane and to represent the graph by a picture. Any such picture is outside the scope of our formal development, but I will draw lots of such pictures informally.

where .

Proof of (5.39). For all ,

Hence, . (Two functions are equal when they have the same domain, the same codomain, and the same rule.)

- is increasing on if for all .
- is strictly increasing on if for all .
- is decreasing on if for all .
- is strictly decreasing on if for all .

- is increasing on if for all .
- is decreasing on if for all .

The proof is by induction, and is omitted.

Proof: The proof follows from induction on or by factoring ,
and is omitted.

Proof: First I will construct a binary search sequence in
such
that

By completeness of , I'll have for some . I'll show , and the proof will be complete.

Let
. Then

For , define

The proof that is a binary search sequence and that for all is the same as the proof given in example 5.16 for , and will not be repeated here. By completeness for some . Since , we have . It follows that

By the formula for factoring (cf. (3.78)), we have

for all . By Archimedean property 3 (cf corollary 5.28), it follows that , i.e .

Let . Since is strictly increasing on , it follows from exercise 5.48A that has at most one solution in and this completes the proof of the theorem.

Proof: [of part b)]
Let
where are integers and are
positive integers. Then (by laws of exponents for integer exponents),

Also,

Hence, , and hence by uniqueness of roots.

one is in , and the other is not in .

the following lemma is assumed: that the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area. The earlier geometers have also used this lemma.[2, p 234]

Euclid indicated that his arguments needed the Archimedean property by using the following definition:

Magnitudes are said toHere ``multiplied'' means ``added to itself some number of times'', i.e. ``multiplied by some positive integer''.have a ratioto one another which are capable, when multiplied, of exceeding one another.[19, vol 2, p114]

Rational exponents were introduced by Newton in 1676.

Since algebraists write etc., for etc., so I write for and I write etc. for etc.[14, vol 1, p355]Here denotes the cube root of .

Buck's *Advanced Calculus*[12, appendix 2] gives eight
different characterizations of the completeness axiom
and discusses
the relations between them.

The term *completeness* is a twentieth century
term. Older books speak about the *continuity*
of the real numbers to describe what we call completeness.