Homework for MAT 603 with Pugh s Real Mathematical Analysis Damien Pitman
CHAPTER 1 Real Numbers 1. Preliminaries (1) In what sense is Euclid s method of reasoning superior to Aristotle s? (2) What role does analogy play in mathematics? Ch. 1: 1, 2(b,c,d), 5(a). 2. Cuts (1) How are real numbers related to Dedekind cuts? (2) What does x < y mean if x and y are cuts? (3) How do Q and R differ with respect to l.u.b.s? (4) What is trichotomy? (5) What must be true of A R if sup A = l.u.b. A? (6) Describe two ways in which R is complete? (7) For sequences, how is satisfying the Cauchy criterion different from satisfying the ɛ N definition of convergence? Why can we functionally ignore any conceptual differences between a Cauchy sequence and a convergent sequence? (8) For each sequence below either provide an ɛ N proof that the sequence converges to its limit or else show that it does not satisfy the Cauchy condition. a n = 1 2( 1) n c n = n 1 n 3 + 2 e n = cos(πn/3) (9) Guess the limit. Ch. 1: 11, 13 b n = 2 + ( 1/3) n d n = 5n + 4 3n + 2 g n = cos n (n) h n = n n 2 1 i n = ( 1)n + ( 2) n+1 3 n j n = nπ e n 3. Euclidean Space 3
4 1. REAL NUMBERS (1) Let x = ( 1, 2), y = (1, 0), and z = (0, 2) be elements of the Euclidean space E with the inner product,. (a) What is the common symbol for E? (b) Find the inner product x, y. (c) Verify symmetry of the inner product with x and y. (d) Verify bilinearity of the inner product with x, y, and z. (e) Verify the Cauchy-Schwartz inequality with x and y. (f) Verity the triangle inequality with x and y and sketch the relevant triangle in the plane. (g) Verify the triangle inequality for distance with x, y, and z and sketch the relevant triangle in the plane. (2) Why is the triangle inequality called the triangle inequality? (3) Is the Cauchy-Schwartz inequality strict in C([0, 2π], R) with f (x) = sin x and g(x) = cos x? (4) Based on the inner product in the last problem, does it make sense to say that sine and cosine are orthogonal in C([0, 2π], R)? Ch. 1: 18, 19, 26, 28(a) 4. Cardinality (1) Is every infinite subset of a countable set denumerable? (2) Is every infinite proper subset of an uncountable set denumerable? (3) Find a function from N to N that is surjective, but not injective. (4) Find a linear function that verifies the equicardinality of any two intervals that do have the same cardinality? Ch. 1: 36(a) 5. Comparing Cardinalities (1) Use the Schroeder-Bernstein theorem to show that [0, 1] [2, 3] has the same cardinality as R. Ch. 1: 33 or 34 Note for 33: By the Fundamental Theorem of Continuous Functions on pg. 39, there is no continuous bijection from [a, b] to R. 6. The Skeleton of Calculus (1) Could Theorem 22 reasonably be summarized by saying that continuous functions are bounded on closed intervals? (2) Could Theorem 23 reasonably be summarized by saying that continuous functions attain extreme values on closed intervals? (3) Consider the IVT as stated in the book. Find two values of c for γ = 1 if f (x) = 2 cos(x) on the interval [0, 2π] (4) Show that f (x) = ex π is continuous at π/e. Ch. 1: 39 (also show that f (x) = x 2 is continuous at every real number), 40, 42
CHAPTER 2 A Taste of Topology 1. Metric Space Concepts (1) Is it true that every sequence of real numbers is a metric space with absolute value as the inherited metric? (2) Give an example where a function f is not continuous, a function g is continuous, and f g is continuous. (3) Consider B = M 1 ((0, 0)) A where A is the set of points on the unit circle with radian measure that is a rational multiple of π. Show that B is neither open nor closed. Also show that B B by giving an example of a point in B \ B. (4) Find a set that has infinitely many cluster points, but does not condense at any point. Hint: Don t try too hard. There is a very well known set that answers this question. (5) Use set-builder notation to describe a subset of R m that is neither open nor closed. Ch 2: 1, 3-7, 9, 11, 26, 29, 30, 89(a,e) 2. Compactness (1) How are Theorems 29 and 36 related? (2) How does Theorem 43 relate to problem 40 from Chapter 1? Ch 2: 38, 84 5
CHAPTER 3 Functions of a Real Variable 1. Differentiation (1) Write up the precise definition of what it means for a function to have a limit at a point. (2) Consider the behavior of each of the functions f (x) = x 2 3, g(x) = (x 1) 1, and h(x) = 3x + 1 at x = 1. Use the definition of the limit to prove that a limit exists or that the limit does not exist. (3) What do f and x mean, and how do they relate to the difference quotient in the definition of differentiability? How does the difference quotient relate to the slope of a secant line? (4) What is it that the derivative of a function evaluates? (5) Provide a more thorough explanation of why Corollary 2 is true. (6) Use the definition of the derivative to prove that each of the functions f (x) = x 2 3, g(x) = (x 1) 1, and h(x) = 3x + 1 is differentiable on any open interval in its domain. Then find the point in the domain of g where g is not differentiable and prove what you have discovered. (7) Prove that f (x) = x 2 is continuous at x = 0, but is not differentiable there. (8) Investigate the continuity and differentiability of the bijection in Exercise 34 of Chapter 1. (9) Find any values of θ from the mean value theorem for f (x) = x 1/3 on the interval [ 1, 8]. (10) Use L Hospital s rule to evaluate each of the following limits: lim x π sin x x π on (0, π) and lim x 1(x 2 1) ln(1 x) on (0, 1). Take-home problem for the final exam [ ] (1) For n = 0, 1, 2, 3,..., let a n = 4+2( 1) n n. 5 (a) Find lim sup n a n, lim inf n a n, lim sup a n+1, a a n and lim inf n+1. a n (b) Does the series n=0 a n converge? (c) Find the interval of convergence for the power series n=0 a nx n. 7