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Reading: • For this assignment, Chapter 3. • For Monday, September 25: Section 3.2, 3.3. 3.4 (Subsequences, Cauchy Sequences, Upper and Lower Limits) • For Wednesday, September 20: Rest of Chapter 3 (Some special sequences, Series, Root and Ratio Tests, Summation by Parts, Absolute Convergence, Rearrangements). Definitions: Subsequences (in a metric space), limit supremum, limit infimum. Exercises to turn in: Problem 1. Rudin 2.19 Problem 2.

PN n Given an integer N, and an a ∈ {0, 1}N , let xa = n=1 2a , and let Ja = 3n [xa , xa +3−N ]. Let K be the middle-thirds cantor set in R constructed as in Example 2.44 of Rudin (in the section on Perfect sets). (a) Prove that for natural numbers N ≤ M, and a ∈ {0, 1}N , b ∈ {0, 1}M we have that Ja ⊇ Jb if aj = bj for all 1 ≤ j ≤ N, and that Ja ∩ Jb = ∅ if there is some 1 ≤ j ≤ N so that aj 6= bj . (b) Let EN be the set constructed in the N th stage of the construction of Cantor set. Prove that [ Ja , EN = a∈{0,1}N

and that this is a disjoint union (i.e. that Ja ∩ Jb = ∅ if a, b ∈ {0, 1}N and a 6= b). N (c) Prove that x ∈ K if and only if there is a sequence a = (an )∞ n=1 ∈ {0, 1} so that ∞ X 2an x= . 3n n=1

(Recall that {0, 1}N denotes the space of {0, 1} sequences). You are allowed to use basic facts about convergence of sequence and series on the real line. (d) Using part (c), construct an explicit bijection of K with {0, 1}N , and use this to show that K is uncountable. You must prove that the function you construct is a bijection. Problem 3. Let (X, d) be a metric space. (a) Prove that if E, F are connected subsets of X and E ∩ F 6= ∅, then E ∪ F is connected. (b) Let E ⊆ X. Define a relation ∼ on E by saying that x ∼ y for x, y ∈ E if and only if there is a nonempty, connected subset F of X so that x, y ∈ F. Prove that this is an equivalence relation, i.e. that it satisfies the following three properties: • If x, y, z ∈ X and x ∼ y and y ∼ z, then x ∼ z. • for all x ∈ X we have x ∼ x, 1

2

• if x, y ∈ X and x ∼ y, then y ∼ x. (Note: for each x ∈ E, the equivalence class [x] = {y ∈ E : y ∼ x} is called the connected component of x in E, the set of equivalence classes of this equivalence relation are called the connected components of E.) (c) We say that (X, d) is locally connected if for every x ∈ X, and every open W ⊆ X with x ∈ W, there is an connected, open subset V of X with x ∈ V, and V ⊆ W. Suppose that (X, d) is locally connected, and let U ⊆ X be open in X. Prove that each connected component of U is open in X. Problem 4. Rudin 2.29. Hint: It may be helpful to use the proceeding problem, the fact that an equivalence relation on a set partitions the set into equivalence classes, and our characterization of the connected subsets of R from class. Problem 5. Rudin 3.7 Problem 6. Rudin 3.16 Problem 7.

P∞ (a) Let X = RN equipped with the metric d(x, y) = n=1 min(1, |xn −yn |)2−n (you are allowed to assume that this is indeed a metric). Let x(n) be a sequence in X, and let x ∈ X. Prove that x(n) →n→∞ x with respect to this metric d if and (n) only if for every m ∈ N we have x m →n→∞ xm . ∞ (b) Regard ℓ (N) as the bounded sequences in RN , and let d ∞ be the ℓ∞ -metric defined on ℓ∞ (N) by d ∞ (x, y) = sup |xn − yn |. n

Given an explicit example of a sequence x(n) ∈ ℓ∞ (N) and an x ∈ ℓ∞ (N) so that d(x(n) , x) → 0 for the metric d in part (a), but so that x(n) does not converge with respect to the metric d ∞ . Problem 8. Let (X, d) be a bounded metric space and F (X) its set of nonempty closed subsets. Recall that the Hausdorff metric on F (X) is defined by d Haus(E, F ) = inf{r > 0 : E ⊆ Nr (F ), F ⊆ Nr (E)}. ∞ (Kn )n=1

Let be a decreasing sequence of nonempty compact subsets of X. Prove that the sequence ∞ \ lim Kn = Kn n→∞

n=1

in the Hausdorff metric. (It may be helpful to use that Nr (A) is open for all r > 0 and all A ⊆ X.)...