Download MAT 300/371 Mathematical Structures/Advanced Calculus (Why) is

MAT 300/371
Mathematical Structures/Advanced Calculus
(Why) is there a number whose square in 2 ?
Theorem: The square of every rational number is different from 2.
To check your proof, successively replace 2 by 3, 4, 6 and 180 and see if the argument still goes through.
Recall the Bernoulli inequality: For h > −1 and n ∈ Z+ , (1 + h)n ≥ 1 + nh (easy proof by induction)
Recall the arithmetic-geometric-harmonic mean inequalities: If a, b ∈ R+ then
√
a+b
2
≥
ab ≥ 1 1
2
a + b
These inequalities are useful in lots of settings, and may also help in the following exercise. There are
dozens of nice proofs of these inequalities; likely the easiest algebraic ones (and most boring) simply
play around with squares being nonnegative: (a ± b)2 ≥ 0, expanding the square and rearranging the
terms. A simple calculus argument factors out an a, sets x = ab , and analyzes the second derivatives
√
2x
of the functions x 7→ 12 (1 + x), x 7→ x, and x 7→ 1+x
at x = 1.
Exercise: Consider the sequence defined recursively by a1 = 1 and
an+1 =
an +
2
2
an
• Show that if a2n < 2 then a2n+1 > 2.
• Show that if a2n > 2 then a2n+1 > 2.
• Show that if a2n < 2 then an+1 > an .
• Show that if a2n > 2 then an+1 < an .
• Show that if L ∈ Q is a lower bound for the set {an : n > 1} then L2 < 2.
1 n
1 n+1
and bn = 1 +
Exercise: Consider the sequences defined by an = 1 +
n
n
• Show that if a is strictly increasing. (Hint: Use the Bernoulli inequality after factoring the ratio
(an /an−1 ) – one of the two choices of the exponent works MUCH better than the other!)
• Show that if b is strictly decreasing.
• Show that if a bounded.
• Show that if b bounded.
• Show that for all m, n > N then am > bn .
• Show that for every number ε > 0 there exits N ∈ N such that if n > N then an − bn < ε.
Are you now ready for the supremum-axiom?
Fun exercise, small subset: Let {ri : i ∈ N} be an enumeration of the rationals in the interval [0, 1].
i
i
Let ε > 0, and consider the set S = ∪∞
n=1 (ri − 2 ε, ri + 2 ε). Show that S contains all rationals in [0, 1],
and the combined lengths of all intervals in this union is ε, and since these intervals are expected to
possibly overlap the size of the set S is even smaller than ε.
Fun exercise, medium size subset: Look up the (middle-one-thirds) Cantor-set on the WWW.
A nice starting point is http://mathworld.wolfram.com/CantorSet.html.
Show that the endpoints of any removed intervals (assuming open intervals are removed) are never
removed, and show that the number of these endpoints is countable. However, the Cantor set is
uncountable. Challenge: Explicitly determine some elements of the Cantor sets that are not endpoints
of any removed intervals. MAT 371: Show that the Cantor set is a perfect set: It is closed and every
point is a limit point.
Even more fun are variations of this set that do not remove the middle thirds at each stage, but remove
variable fractions of the remaining intervals. To make things worse, one may even take countable unions
of such sets to obtain strange sets of full measure.
The message: Do not underestimate the real numbers. Some of this is subject of MAT 570.