Studying Rudin's Principles of Mathematical Analysis Through ...

14 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS
1.9.27 Theorem
Prove that for any cut α, p ∈ α if and only if p* < α.
1.10 Real Numbers (2ed)
1.10.1 Definition
1.10.2 Theorem (Dedekind)
Let A and B be sets of real numbers such that
(a) every real number is either in A or in B;
(b) no real number is in A and in B;
(c) neither A nor B is empty;
(d) if α ∈ A, and β ∈ B, then α < β. Prove that there is one (and only one) real number γ such that
α ≤ γ for all α ∈ A, and γ ∈ B for all β ∈ B. Corollary Under the hypotheses of Theorem 1.10.2,
either A contains a largest number or B contains a smallest.
1.10.3 Definition
Let E be a set of real numbers. What are upper and lower bounds of E.
1.10.4 Definition
Let E be bounded above. What the least upper bound and greatest lower bounds of E?
1.10.5 Examples
Let E consist of all numbers 1/n, n=1, 2, 3, What are the least upper and greatest lower bounds of E?
1.10.6 Theorem
Let E be a nonempty set of real numbers which is bounded above. The the lub of E exists.
1.10.7 Theorem
For every real x > 0 and every integer n > 0, there is one and only one real y > 0 such that y n = x. This
number y is written n√ x, or x 1/n .
1.11 Exercises
Now, you should be ready to tackle the exercises.