Mathematics 242 – Principles of Analysis Discussion 2 – sup(S) continued September 29, 2004

Background

Last time, we introduced the supremum or least upper bound of a subset S ⊆ R. Recall, we say that s = sup(S) if

• s is an upper bound of S – x ≤ s is true for all x ∈ S, and

• if s′ < s, then there exists x ∈ S such that x > s′.

The completeness axiom, the crucial property of the real numbers, is:

Completeness Axiom. If S is nonempty and bounded above in R, then sup(S) exists in R.

That is, every nonempty set that is bounded above in R has a least upper bound in R. Today, we want to study further properties of the supremum.

Discussion Questions

A) The second defining property of the supremum s = sup(S) above says: if s′ < s, then there exists x ∈ S such that x > s′. Show that this statement is equivalent to saying: If t is any upper bound for S, then t ≥ s. (Note and Hint: In fact, many analysis books define the supremum using this alternate form, because it clearly shows that sup(S) is the “least upper bound” for S. The relation between these statements comes from one of our basic logical equivalences!)

B) Let S be a nonempty bounded subset of R. Show that s = sup(S) is unique. Method: Assume that two real numbers s, s′ both satisfy the definition, then deduce that s must equal s′.

C) Prove that if x < y are any real numbers, then there exist infinitely many rational numbers r with x < r < y. Hint: Last time we showed that there is one such rational number. Try to extend that proof.

D) Let a/b be a rational number written as a fraction in lowest terms with 0 < a/b < 1.

1) Show using the Archimedean Property that there exists an integer n such that

\frac{1}{n + 1} ≤ \frac{a}{b} ≤ \frac{1}{n}.

2) If n is chosen as in part 1, show that a/b − 1/(n + 1) is a fraction that when written in lowest terms has numerator is less than a.

3) Use the Principle of Strong Induction from Problem Set 3 to show that every rational number a/b as above can be written as a sum:

\frac{a}{b} = \frac{1}{n_{1}} + ⋯ + \frac{1}{n_{k}}

for some distinct natural numbers n_{i}. (For instance 49/90 = 1/3 + 1/9 + 1/10.)

Such fractions are often called “Egyptian fractions” because the ancient Egyptians did all of their rational number arithmetic using them (for some reason they didn’t like fractions with a numerator different from 1). They wrote all fractions this way, then computed sums and products via tabulations that have survived in ancient papyrus manuscripts – the “Rhind papyrus” is probably the most famous example. It is one of the oldest surviving ancient mathematical documents. There is an interesting discussion of this online at

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Egyptian papyri.html

Assignment

Group writeups due by 5:00pm on Monday, October 4.