Advanced Calculus Homework # 2 (due 10/16)
1. Let xn be a monotone increasing sequence such that xn+1 − xn ≤ 1/n.
Must xn converge?
2. Let F be an ordered field in which every strictly monotone increasing sequence bounded above converges. Prove that F is complete.
3. Let A ⊂ R and B ⊂ R be bounded below and define A + B = {x + y|x ∈ A and y ∈ B}. Is it true that inf(A + B) = inf A + inf B?
4. Let S ⊂ [0, 1] consist of all infinite decimal expansions x = 0.a1a2a3· · · where all but finitely many digits are 5 or 6. Find sup S.
5. Let xn be a sequence with lim sup xn = b ∈ R and lim inf xn = a ∈ R.
Show that xn has a subsequences un and `n with un → b and `n → a.
6. Let A and B be two nonempty sets of real numbers with the property that x ≤ y for all x ∈ A and y ∈ B. Show that there exists a number c ∈ R such that x ≤ c ≤ y for all x ∈ A and y ∈ B. Give a counterexample to this statement for rational numbers.
7. For nonempty sets A, B ⊂ R, determine which of the following statements are true. Prove the true statements and give a counterexample for those that are false:
a. sup(A ∩ B) ≤ inf{sup(A), sup(B)}.
b. sup(A ∩ B) = inf{sup(A), sup(B)}.
c. sup(A ∪ B) ≥ sup{sup(A), sup(B)}.
d. sup(A ∪ B) = sup{sup(A), sup(B)}.
8. Let xn be a sequence in R such that d(xn, xn+1) ≤ d(xn−1, dxn)/2. Show that xn is a Cauchy sequence.
9. Let P ⊂ R be a set such that x ≥ 0 for all x ∈ P and for each integer k there is an xk ∈ P such that kxk≤ 1. Prove that 0 = inf(P ).
10. Assume that A = {am,n| m = 1, 2, 3, · · · and n = 1, 2, 3, · · · } is a bounded set and that am,n ≥ ap,q whenever m ≥ p and n ≥ q. Show that
n→∞lim an,n = sup A.
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