12 - Additional Theorems, Problems, and Proofs

Summary

In our final chapter, we present a collection of theorems and problems from various branches of mathematics and their proofs and solutions. We begin by discussing some set theoretic results concerning infinite sets, including the Cantor-Schröder-Bernstein theorem. In the next two sections we present proofs of the Cauchy-Schwarz inequality and the AM-GM inequality for sets of size n. We then use origami to solve the classical problems of trisecting angles and doubling cubes, followed by a proof that the Peaucellier-Lipkin linkage draws a straight line. We then look at several gems from the theory of functional equations and inequalities. In the final sections we conclude with an infinite series and an infinite product for simple expressions involving π, and illustrate each with an application.

Denumerable and nondenumerble sets

The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.

David Hilbert

To infinity, and beyond!

Buzz Lightyear, Toy Story (1995)

Two sets have the same cardinality if there exists a one-to-one function from one set onto the other, i.e., a one-to-one correspondence between the sets. An infinite set of numbers is denumerable (or countably infinite) if it has the same cardinality as the set ℕ = {1, 2, 3, …} of natural (or counting) numbers.