Lista - Análise Matemática 2023-1

Section 1.1 The Real Number System 9 It is not useful to define ∞ − ∞, 0 · ∞, ∞/∞, and 0/0. They are called indeterminate forms, and left undefined. You probably studied indeterminate forms in calculus; we will look at them more carefully in Section 2.4. 1.1 Exercises Write the following expressions in equivalent forms not involving absolute values. (a) a + b + |a − b| (b) a + b − |a − b| (c) a + b + 2c + |a − b| − |a + b − 2c + |a − b|| (d) a + b + 2c − |a − b| − |a + b − 2c − |a − b|| Verify that the set consisting of two members, 0 and 1, with operations defined by Eqns. (1.1.11) and (1.1.12), is a field. Then show that it is impossible to define an order on this field that has properties (F), (G), and (H). 3. Show that √2 is irrational. Hint: Show that if √2 = m/n, where m and n are integers, then both m and n must be even. Obtain a contradiction from this. 4. Show that √p is irrational if p is prime. 5. Find the supremum and infimum of each S. State whether they are in S. (a) S = { x | x = (−1)^n(n)! + |1 + (−1)^n|, n >= 1 } (b) S = { x | x^2 < 9 } (c) S = { x | x^2 <= 7 } (d) S = { x | −1 < x + |1 | <= 5 } (e) S = { x − 1/x | x = rational and x^2 < 7 } 6. Prove Theorem 1.1.8. Hint: The set T = { x | − x ∈ S } is bounded above if S is bounded below. Apply property (I) and Theorem 1.1.3 to T. 7. (a) Show that inf S <= sup S (A) for any nonempty set S of real numbers, and give necessary and sufficient conditions for equality. (b) Show that if S is unbounded then (A) holds if it is interpreted according to Eqns. (1.1.12) and the definitions of Eqns. (1.1.13) and (1.1.14). 8. Let S and T be nonempty sets of real numbers such that every real number is in S or T and if S ∈ S and t ∈ T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T. (A decomposition of the reals into two sets with these properties is a Dedekind cut. This is known as Dedekind's theorem.) 10 Chapter 1 The Real Numbers 9. Using properties (A)-(H) of the real numbers and taking Dedekind's theorem (Exercise 1.1.8) as given, show that every nonempty set U of real numbers that is bounded above has a supremum. Hint: Let T be the set of upper bounds of U and S be the set of real numbers that are not upper bounds of U. 10. Let S and T be nonempty sets of real numbers and define S + T = { s + t | s ∈ S, t ∈ T }. (a) Show that sup(S + T) = sup S + sup T (A) if S and T are bounded above and inf(S + T) = inf S + inf T (B) if S and T are bounded below. (b) Show that if they are properly interpreted in the extended reals, then (A) and (B) hold if S and T are arbitrary nonempty sets of real numbers. 11. Let S and T be nonempty sets of real numbers and define S − T = { s − t | s ∈ S, t ∈ T }. (a) Show that if S and T are bounded, then sup(S − T) = sup S − inf T (A) and inf(S − T) = inf S − sup T (B) (b) Show that if they are properly interpreted in the extended reals, then (A) and (B) hold if S and T are arbitrary nonempty sets of real numbers. 12. Let S be a bounded nonempty set of real numbers, and let a and b be fixed real numbers. Define T = {as + b | s ∈ S}. Find formulas for sup T and inf T in terms of sup S and inf S. Prove your formulas. 1.2 MATHEMATICAL INDUCTION If a flight of stairs is designed so that falling off any step inevitably leads to falling off the next, then falling off the first step is a sure way to end up at the bottom. Crudely expressed, this is the essence of the principle of mathematical induction: If the truth of a statement depending on a given integer n implies the truth of the corresponding statement with n replaced by n + 1, then the statement is true for all positive integers n if it is true for n = 1. Although you have probably studied this principle before, it is so important that it merits careful review here. Peano's Postulates and Induction The rigorous construction of the real number system starts with a set N of undefined ele- ments called natural numbers, with the following properties.