begin quote ita"
CHAPTER   II   [:]   The Real Number System "end quote
topic index     contents

CHAPTER   II   PREVIEW

Since introductions are always written last, this space will be completed after my reading is finished.

§1 The Field Properties

See the statements of the field properties on the spoken reals page beginning here.

The experience of my two previous readings has led me to skip F1 and F2 on ita page 17. These are the first two consequences of the field properties. I want to see if I can get through the chapter without them, and I would like to prove them after we have the positive integers.

§2 The Order Property                  Back to reading:     II     §1     F10

See the statement of the order property on the spoken reals page here. With additional concepts below, here is Rosenlicht's statement of the order property:

begin quote ita-2-2-1-pg19 "
The order property of the real number system is the following:

PROPERTY VI.    There is a subset of   R   such that
(1) if   a, b ,   then   a+b, a·b
(2) for any   aR,   one and only one of the following statements is true
a
a
= 0
-a.

The elements   aR   such that   a   will of course be called positive, those such that   -a   negative. From the above property of     we shall deduce all the usual rules for working with inequalities.
"end quote

Here also is Rosenlicht's introduction of the inequality symbols:

begin quote ita-2-2-3-pg19 "
To be able to express the consequences of Property VI most conveniently we introduce the relations   ""   and   "".   For   a, bR,   either of the expressions   ab   or   ba   (read respectively as   "a is greater than b"   and "b is less than a") will mean that   a-b. Either of the expressions   ab   or   ba   will mean that   ab   or   a=b.

Clearly   a   if and only if   a0.   An element   aR   is negative   if and only if   a0.
"end quote

§3 The Least Upper Bound Property                  Back to reading:     II     §2     O9

See the statement of the least upper bound property on the spoken reals page here, and see the general description from my second reading here. In the first reading web pages, I did not include any of this general material. In both the first and second readings, I missed an important and very interesting point, which I will present after Rosenlicht's introduction:

begin quote ita-2-3-1-pg23 "
To introduce the last fundamental property of the real number system we need the following concepts. If   S   then an upper bound for the set S is a number   a   such that   sa   for each   sS.   If the set   S   has an upper bound, we say that   S   is bounded from above. We call a real number   y   a least upper bound of the set S if   (1)   y   is an upper bound for   S   and   (2)   if   a   is any upper bound for   S,   then   ya.
"end quote

In my second reading, I missed Rosenlicht's statement, on page 25, that any real number is an upper bound for the empty set. To me, this does not make sense if we focus on the definitions just presented. However, any real number ... that is greater than any given upper bound for a set of real numbers ... is not in the set. Since every real number is not in the empty set, it seems to make sense to say that any real number is an upper bound for the empty set. Let's see about this:

Let   a   be a real number such that every real number,   x,   which is greater than   a   is not in a given set   S   of real numbers. I will try to show that   a   is an upper bound on   S.   By trichotomy, every element of   S   is either less than or equal to   a   or greater than   a.   But by hypothesis, every number greater than   a   is not in   S.   This means by trichotomy that every element of   S   must be less than or equal to   a,   and we see therefore that   a   is an upper bound on   S.

(1) The least upper bound is unique.

begin quote ita-2-3-2-pg23 "
From [the definition of the least upper bound] it follows that two least upper bounds of a set   S   must be less than or equal to each other, hence equal. Thus a set   S   can have at most one least upper bound and we may speak of the least upper bound of   S   (if one exists).
"end quote

(2) Any real number less than the least upper bound is less than some element of the set.

begin quote ita-2-3-2-pg23 "
Note also the following important fact: if   y   is the least upper bound of   S   and   x,   xy,   then there exists an element   sS   such that   xs.
"end quote

(3) Example:   nonempty   finite   subsets   of the set of real numbers

begin quote ita-2-3-3-pg24 "
A nonempty finite subset   S   always has a least upper bound; in this case the least upper bound is simply the greatest element of   S.
"end quote

It is the case that if a nonempty set of real numbers has no greatest element, then it must be infinite?

(4) Example:   max S

begin quote ita-2-3-3-pg24 "
More generally any subset   S   that has a greatest element (usually denoted   max S)   has   max S   as a least upper bound.
"end quote

(5) Example:   infinite subsets of the set of real numbers

begin quote ita-2-3-3-pg24 "
But an infinite subset of     need not have a least upper bound, for example     itself has no upper bound at all.
"end quote

(6) Example:   The least upper bound is in the set?

begin quote ita-2-3-3-pg24 "
Furthermore, if a subset   S   of     has a least upper bound, it does not necessarily follow that this least upper bound is in   S;   for example, if   S   is the set of all negative numbers then   S   has no greatest element, but any   a0   is an upper bound of   S   and zero (a number not in   S)   is the least upper bound of   S.
"end quote

(7) and, not finally, property vii:

begin quote ita-2-3-4-pg24 "

PROPERTY VII. ( LEAST UPPER BOUND PROPERTY ).

A nonempty set of real numbers that is bounded from above has a least upper bound.
"end quote

(8) notation and converse

begin quote ita-2-3-7-pg25 "
The least upper bound of a subset   S   of     will be denoted l.u.b S; another common notation is sup S (sup standing for the Latin supremum). Property VII says that l.u.b S exists whenever   S   is nonempty and bounded from above. Conversely, if   S   and l.u.b. S exists, then   S   must be nonempty (for any real number is an upper bound for the empty set and there is no least real number) and bounded from above.
"end quote

(9) the greatest lower bound

begin quote ita-2-3-8-pg25 "
Analogous to the above there are the notions of a lower bound and greatest lower bound:   a   is a lower bound for the subset   S   [of]     if   as   for each   sS,   and   a   is a   greatest lower bound of S   if   a   is a lower bound of   S   and there exists no larger one.   S   is called bounded from below if it has a lower bound.
"end quote

(10) deductions

begin quote ita-2-3-8-pg25 "
It follows from Property VII that every set   S   of real numbers that is nonempty and bounded from below has a greatest lower bound:

as a matter of fact,

[1] a set   S   is bounded from below if and only if the set   S={x: -xS}   is bounded from above, and

[2] if   S   is nonempty and bounded from below then   -l.u.b S   is the greatest lower bound of S.
"end quote

Recall please the two parts of an if and only if statement.

The "if" part of [1] is as follows:   if   the set   S={x: -xS}   is bounded from above,   then   S   is bounded from below. Let   a   be an upper bound on   S. Then for all   xS   we have   xa.   Applying O3 twice, this becomes   -a-x   which shows that   -a   is a lower bound on the set S.

The "only if" part of [1] is as follows:   if   a set   S   is bounded from below,   then   the set   S={x: -xS}   is bounded from above. Let   a   be a lower bound on   S.   Then for all   xS   we have   ax.   Applying O3 twice, this becomes   -x-a.   But   -(-x)=x   (by F6) so that as   x   ranges over all of   S,   -x   ranges over all of   S   and we see consequently that   S   is bounded from above.

To prove [2], note from [1] that   S is bounded from above and both sets are nonempty so that   S has a least upper bound. If   a   is any lower bound on   S   then   -a   is an upper bound on   S, and we must have   l.u.b.S-a.   Applying O3 twice, this becomes   a-l.u.b.S   and we have the desired result.

(11) notation and min S

begin quote ita-2-3-8-pg25 "
The greatest lower bound of a subset   S   of     is denoted   g.l.b. S;   another notation is   inf S   (inf abbreviating the Latin infimum). If   S   has a smallest element (for example, if   S   is finite and nonempty) then   g.l.b. S   is simply this smallest element, often denoted   min S.
"end quote

(12) certain gaps

begin quote ita-2-3-9-pg25 "
We proceed to draw some consequences of Property VII. Among other things we shall show that the real numbers are not very far from the rational numbers, in the sense that any real number may be "approximated as closely as we wish" by rational numbers. The way to view the situation is that the rational numbers are in many ways very nice, but there are certain "gaps" among them that may prevent us from doing all the things we would like to do with numbers, such as solving equations (e.g., extracting roots), or measuring geometric objects, and the introduction of the real numbers that are not rational amounts to closing the gaps.
"end quote

On to §3: LUB 1