Bridge to Abstract Mathematics: Mathematical Proof and Structures

318 PROPERTIES OF NUMBER SYSTEMS Chapter 9

Also, since I is not bounded below, there exists d E I such that d < x.

Hence d < x < c, d E I, and c E I. Since I is an interval, we have x E I,

so that (- oo, b] s I, and (- GO, b] = I, as desired.

In Exercise 8 you should handle the subcase b & I, as well as the re-

maining three cases described at the start of the proof.

The preceding material should suffice to convince you of the importance

of the completeness property of R and especially of its relevance to material

encountered in previous calculus courses. The least upper bound property

has additional important applications to calculus, the most familiar one

probably being the so-called extreme value theorem: A function continuous

on a closed and bounded interval attains an absolute maximum and mini-

mum value on that interval. This result is the basis for part of the approach

to applied "max-min" problems taught in most first-semester calculus

courses. Its proof is based on a very technical property of R known as the

Heine-Bore1 theorem. The latter, in turn, follows, in a nontrivial fashion,

from the completeness property of R. The details, including the proof, of the

Heine-Bore1 theorem and its consequences are a standard part of a course

in advanced calculus; we omit them from this text.

Exercises

1. Use the definitions of lub and glb in the ordered field R to prove:
   (a) b = lub [a, b], where a, b E R, a < b
   (b) a = glb (a, a), where a E R
   (c) 1 =lub{l/nln~N); O=glb{l/n(n~N)
   (d) 1 = glbN
   *(e)^3 = glb ((3,4) u (6));^6 = lub ((3,4) u (6))


2. Prove that a subset S of an ordered field F is bounded in F if and only if it is
   bounded both above and below in F.


3. (a) Formulate a definition of 1 = glb, S, analogous to Definition 3.
   (b) Prove that a least upper bound of a subset of an ordered field F, if it exists,
   is unique.


4. (a) Suppose that a subset S of an ordered field F is not bounded above in F.
   Let T be a subset of F satisfying the property that, for each x E S, there exists
   y E T such that x < y. Piove that T is not bounded above in F.
   (b) Suppose S and T are subsets of a complete ordered field F, both bounded
   above in F, satisfying the property that for all x E S, there exists y E T such that
   x < y. Prove lub S I lub T.
   *(c) Suppose S and T are subsets of a complete ordered field F, both bounded
   above in F, such that S G T. Prove that lub S < lub T.


5. Let S be a subset of an ordered field F such that u = lub S.
   (a) Prove that -u = glb T, where T = { - x, x E S).
   (b) Prove that if v E F and v is an upper bound for S, then u I v.