At 4:36 PM 1/19/1997, Jim Choate wrote:
"The least upper bound of a set of real numbers is often called the supremum, the greatest lower bound its infimum. In general the supremum and infimum of a set ARE MEMBERS OF THE SET or at least LIMITS OF SEQUENCES OF MEMBERS OF THE SET."
[capitalization is mine]
In this case the suprenum is infinity.
Introduction to Calculus and Analysis Courant and John Vol. 1, pp. 97, Section e. 1965 Edition Library of Congress: 65-16403
I presume this is a reference to Jim's earlier post: At 1:19 PM 1/19/1997, Jim Choate wrote:
Forwarded message:
Date: Sun, 19 Jan 1997 12:20:45 -0600 From: Sir Robin of Locksley <tozser@stolaf.edu>
Is it possible to prove that number 0.1234567891011121314151617181920... iz irrational?
Most definately. All you need to do is prove that the set of this number is uncountable, ergo is irrational. If you have friend who have done math in real analysis they can explain more.
irrational means 'not expressible as the ratio of two integers', this does not imply uncountable.
Whether 0.12345678910111213... is irrational or not depends on how we choose to define a rational number where the denominator and numerator are both approaching infinity and how quickly those approaches occur.
[ What is your definition of infinity/infinity ?]
As alluded to before,
0.12345678910111213 = 12345678910111213.... / n
(where n = infinity)
0.123456789101112... is certainly countable because by the definition of countable it must be 1-to-1 with the counting numbers (ie non-negative integers), which this one clearly is since it contains each positive non-zero integer (ie the set of numbers required to produce the number are clearly less, 1 less to be exact, than the counting numbers).
Of course this same argument could be used to claim that pi is also rational. pi = 314159.... / n (where n=infinity). What could be the problem here? One problem might be that an integer (not including infinity) divided by "infinity" should be 0. The p/q definition of rational numbers does not rely on "infinity" being a member of the set of integers. Also, we rarely hear this dialogue: "Give me an integer." "Okay - infinity!" Still, let's define a set which contains the integers as well as positive and negative "infinity". I will call this set the "Choate-integers." Let's add some reasonable rules for working with +inf and -inf. n is a non-zero positive traditional integer. a. -inf = +inf * -1 b. +inf * n = +inf c. +inf + n = +inf d. +inf / +inf = 1 Operations involving multiplication or division of +inf and zero are undefined. Using these definitions, you will find that the p/q definition of rational numbers operates as expected on the set of Choate-integers. The argument I supplied earlier for the rational nature of 0.1234... still holds. Math Man