Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Homework Schedule

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1 Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Homework Schedule Notations I will use the symbols N, Z, Q and R to denote the set of all natural numbers, the set of all integers, the set of all rational numbers and the set of all real numbers respectively. I will prefer to use these symbols liberally and interchangeably with the correspinding symbols J, Z, Q and R used in Gaughan s book to denote the same sets. I hope this preference of mine does not cause much confusion. I will also use the following language for Additional Problems. A indicates that it is an important problem (either the result itself or the concepts (methods) used in the proof are really important), but is not assigned due to either it being too time-consuming to writeup carefully, or it is discussed in class, or that it is a bit too hard to assign as a homework problem, or some other reason. It is important that you try to get the crux of the arguments for these problems. A # indicates that the problem is not so important in terms of understanding concepts that may follow in the future, but the purpose of such a problem is to give you more practice with similar kinds of problems so that your understanding in that particular concept is strengthened. A $ indicates that the problem is an interesting result from an aesthetic point of view, but it could also be useful in strengthening the understanding (especially, when a nice picture of the underlying concepts is painted by such a problem). Homework 1 Due Date: Thursday, 06/16/ Problem 6 in Chapter 0 of Gaughan s book: If A B, prove that C\B C\A. Either prove the converse is true or give a counterexample. 2. Problem 4 in Chapter 0 of Gaughan s book: Prove the De Morgan s Law: A\(B C) = (A\B) (A\C) 3. Problem 12 in Chapter 0 of Gaughan s book: Use De Morgan s Laws to give a different and simpler description (the description should not involve any infinite unions or intersections) of the following sets: 1

2 (a) (b) R\ n=1 n=1 ( 1 n, 1 ) n [ 1 R\ n, ] n 4. Problem 13 in Chapter 0 of Gaughan s book. 5. Problem 17 in Chapter 0 of Gaughan s book. 6. Let A, B and C be sets and let f : A B and g : B C be functions. Prove that if f and g are injective then so is g f and similarly, if f and g are surjective, then so is g f. However, the respective converses of these statements do not hold (give counterexamples). All you can say is that if g f is injective then f is injective, and similarly, if g f is surjective then g is surjective. Additional Problems (do not turn in): 1. Problem 3 in Chapter 0 of Gaughan s book. (#) 2. Problem 18 in Chapter 0 of Gaughan s book. Note that the result is a very fundamental result for bijective functions. ( ) Homework 2 Due Date: Tuesday, 06/21/ Problem 21 in Chapter 0 of Gaughan s book: Prove that n 3 + 5n is divisible by 6 for each n N. 2. Problem 38 in Chapter 0 of Gaughan s book: Let a, b, c and d be any real numbers such that a < b and c < d. Prove that [a, b] is equivalent to [c, d]. (Hint: Show that [a, b] is equivalent to [0, 1] first.) (Additional hints: You might consider special cases when a = c = 0 and c = d, and then gradually generalize. But then writing down special cases is time consuming. Use the special cases to get the general idea and then just write down a crisp proof. Also recall that compositions of bijective functions are also bijective. This could be useful depending on the way you choose to prove.) 3. Problem 32 in Chapter 0 of Gaughan s book: Let P n be the set of all polynomials of degree n with integer coefficients. Prove that P n is countable. (Hint: A proof by induction is one method of approach.) (Additional Hints/Outline of Proof: There are at least two ways I can think of. Method 1: First use Theorem 0.16 along with mathematical induction to prove that the (n + 1)-fold cartesian product A 0 A 1 A 2... A n is countable whenever each A i is countable. From this deduce that the (n + 1)-fold cartesian product of Z, that is, 2

3 Z Z... Z = Z n+1 is countable. Then construct a (the obvious!) injection from P n to Z n+1. Voila! Method 2: Use induction directly. P 0 is countable (why?). Assuming P k is countable for some k 0, write P k+1 as a countable union of subsets of P k+1 each of which is countable by the induction hypothesis (along with some mild tweaking by constructing appropriate bijections). Then use Exercise 34 which is a generalization of Theorem 0.17 that says that a countable union of countable sets is countable. ) 4. Problem 33 in Chapter 0 of Gaughan s book: Use the previous problem to show that the set of all polynomials with integer coefficients is a countable set. (Hint: Use the fact that a countable union of countable sets is countable.) 5. Assuming that R is uncountable, prove that the set of all irrational numbers is uncountable. Additional Problems (do not turn in): 1. Problem 22 in Chapter 0 of Gaughan s book. (#) 2. Problem 37 in Chapter 0 of Gaughan s book. After proving the given statement, also deduce that card(a) < card(p (A)). That is, there is an injection i : A P (A) but there is no bijection from A onto P (A). ($, ) 3. Problem 36 of Chapter 0 of Gaughan s book: Show that the set of all algebraic numbers are countable. (Hint: First do problems 32, 33, 34 and 35 and finally put them all together in order to prove problem 36.) ($, #) Homework 3 Due Date: Thursday, 06/23/ Problem 44 in Chapter 0 of Gaughan s book: If x = sup S, show that for each ɛ > 0, there is a S such that x ɛ < a x. 2. Problem 43 in Chapter 0 of Gaughan s book: Let A = {r : r is a rational number and r 2 < 2}. Prove that A has no largest member. ( Hint: If r 2 < 2, and r > 0, choose a rational number δ such that 0 < δ < 1 and δ < 2 r2 2r + 1. Then show that (r + δ)2 < 2. Compare with the proof of Theorem ) 3. Find the suprema, infima, maxima and minima of the given sets. If any of those do not exist, say so. { } 1 (a) S = 2n 1 : n N (b) S = {x : x (0, 1) Q} (c) S = { x : x [0, ) Q and x 2 < 2 } 3

4 4. Let A and B be nonempty subsets of the real numbers and suppose that they are both bounded from above. If for every a A, there exists b B such that b a, prove that sup A sup B. 5. Let A be a nonempty bounded subset of R. Define the set A by A := { a : a A}. Prove that inf( A) = sup A. (Comments (think about these, but do not turn them in): Observe how you can generalize the result to find inf(ca) for any negative constant c, where the set ca := {ca : a A}. What about the case when c 0? What about sup(ca)?) 6. Let A and B be nonempty bounded subsets of R. Define the set A + B by A + B := {a + b : a A, b B}. Prove that sup(a + B) = sup A + sup B. (Hint: First let α = sup A and β = sup B and show that α + β is an upper bound for A + B. Then show that it is the least upper bound as follows. Let γ < α + β. Set ɛ = α + β γ. Using Problem 1, show that γ is not an upper bound for A + B. A better hint: Set 2ɛ = α + β γ. Then use Problem 1. An alternate way: To show that α + β is the least upper bound, we might also use the second property of being a supremum more directly (without the need for the epsilonstuff). That is, we can prove in the following way: Let γ be any upper bound for the set A + B. We wish to show that γ α + β. Suppose that γ < α + β. Then γ β < α. Now use the fact that α = sup A. In the same way, use the fact that β = sup B after another similar inequality manipulation. Your goal is to find (a + b) for some a A and for some b B such that γ < (a + b). (Do you see why?) ) Additional Problems (do not turn in): 1. Problem 45 in Chapter 0 of Gaughan s book. This is the analogue of the result in Problem 44 for infimums. As another small exercise before you start with the proof, first write down the statement of the result before looking at the book, but just by looking at the statement for supremum given in problem 44. ( ) 2. Make sure you know how to prove your answers in problem 3. (#) 3. Problems 39, 40, 41 in Chapter 0 of Gaughan s book. It is important that you know how to prove these results from the axioms defining the real numbers. (, $) Homework 4 Due Date: Tuesday, 06/28/ Problem 2 in Chapter 1 of Gaughan s book: Let x and y be distinct real numbers. Prove that there is a neighbourhood P of x and a neighbourhood Q of y such that P Q =. (This is called as the Hausdorff property of the real numbers as a topological space.) 2. Problem 6 in Chapter 1 of Gaughan s book (parts (b), (d)): Use the definition of convergence to prove (carefully!) that each of the following sequences converges (use your calculus knowledge to make guesses for the limits): 4

5 { 2 2n (a) n { 3n (b) 2n + 1 } n=1 } n=1 3. Problem 7 in Chapter 1 of Gaughan s book: Show that {a n } n=1 only if {a n A} n=1 converges to 0. converges to A if and 4. Problem 9 in Chapter 1 of Gaughan s book: Prove the Squeeze Theorem: Suppose {a n } n=1, {b n} n=1 and {c n} n=1 are sequences such that {a n} n=1 converges to A and {b n} n=1 converges to A, and a n c n b n for all n. Prove that {c n } n=1 converges to A. (Additional thoughts (do not turn in): Can you extend the argument and prove the result with the slightly weaker assumption that a n c n b n for all n N 0 for some fixed N 0 N? ) 5. Problem 14 in Chapter 1 of Gaughan s book: Prove that every Cauchy sequence is bounded. (Hint: Argue similarly as you did in proving Theorem 1.2.) 6. Prove directly (do not use Theorem 1.8) that, if {a n } n=1 and {b n} n=1 are Cauchy, then so is {a n + b n } n=1. (Hint: Use the triangle inequality in an ɛ/2-argument.) 7. Problem 22 in Chapter 1 of Gaughan s book (only the statement concerning the supremum): Let S be a nonempty set of real numbers that is bounded from above and let x = sup S. Prove that either x belongs to S or x is an accumulation point of S. Additional Problems (do not turn in): 1. Problem 10 in Chapter 1 of Gaughan s book: Prove that if {a n } n=1 converges to A, then { a n } n=1 converges to A. Is the converse true? Justify your conclusion ( ) 2. Give an example of a sequence that is bounded, but not convergent. (, #) 3. Give an example of a subset of R that has countably many accumulation points. (, #) 4. Problem 17 in Chapter 1 of Gaughan s book: Prove (directly using the definition and without { } using any theorems on Cauchy and convergent sequences) that the sequence 2n + 1 is Cauchy. n n=1 Homework 5 Due Date: Thursday, 06/30/ Problem 27 in Chapter 1 of Gaughan s book: Suppose {a n } n=1 and {b n} n=1 are sequences such that {a n } n=1 converges to A 0 and {a nb n } n=1 converges. Prove that {b n} n=1 converges. (Hint: Observe that the sequence {a n } n=1 is eventually bounded away from zero. And recall that convergence of any sequence depends only on the tail of the sequence, that is, the first finitely many terms can be conveniently ignored ). 5

6 2. Problem 28 in Chapter 1 of Gaughan s book: If {a n } n=1 converges to a with a n 0 for all n, show that { a n } n=1 converges to a. (Hint: If a > 0, then a n a = a n a an + a ). 3. Problem 32 in Chapter 1 of Gaughan s book: Find the limit with the general term as given (please (briefly) justify your computations by quoting the use of the relevant theorems or results from other exercises): ( n 2 ) + 4n (a) n 2 5 ( cos n ) (b) n ( ) sin n 2 (c) n ( ) n (d) n 3 3 ) (e) ( 4 1 n 2 n ( ) n (f) ( 1) n n Problem 36 in Chapter 1 of Gaughan s book: Let {a n } n=1 be a bounded sequence of real numbers. Prove that {a n } n=1 has a convergent subsequence. (Hint: You may want to use the Bolzano-Weierstrass Theorem). Additional Problems (do not turn in): 1. Problem 40 in Chapter 1 of Gaughan s book: Show that the sequence defined by a 1 = 6 and a n = 6 + a n 1 for n > 1 is convergent, and find its limit. 2. Problem 35 in Chapter 1 of Gaughan s book: Let {a n } n=1 be a sequence of real numbers. Suppose x is an accumulation point of {a n : n N}. Show that there is a subsequence of {a n } n=1 that converges to x. 3. Problem 47 in Chapter 1 of Gaughan s book: Suppose that {a n } n=1 that converges to A and that B is an accumulation point of {a n : n N}. Prove that A = B. (Hint: Use the previous problem). 4. Problems 25, 26 and 34 in Chapter 1 of Gaughan s book. (, #) 5. Give an example of a sequence that has three subsequences such that two of them converge to two different real numbers and the third is bounded but not convergent. If your example is a bounded sequence, please modify it so that it now has a fourth subsequence that is unbounded. (#) 6. Problem 39 in Chapter 1 of Gaughan s book: Suppose that {x n } n=1 and {y n} n=1 are two sequences both converging to x 0. Define a new sequence {z n } n=1 as follows: z 2n = x n and z 2n 1 = y n. Prove that {z n } n=1 converges to x 0 as well. (, #) 7. Problem 37 in Chapter 1 of Gaughan s book: Prove that if {a n } n=1 bounded, then {a n } n=1 converges. ( ) is decreasing and 6

7 8. Problems 45 and 46 in Chapter 1 of Gaughan s book. ( ) Homework 6 Due Date: Thursday, 07/07/ Problem 2 in Chapter 2 of Gaughan s book: Define f : ( 2, 0) R by f(x) = 2x 2 + 3x 2. Prove that f has a limit at 2 and find it. x Problem 3 in Chapter 2 of Gaughan s book: Give an example of a function f : (0, 1) R that has a limit at every point of (0, 1) except at (1/2). Use the definition to justify the example. Additional thoughts: (do not turn in) What are all the accumulation points of the domain, (0, 1)? Does your example function have limits at all the other accumulation points of the domain? 3. Problem 5 of Chapter 2 of Gaughan s book: (Uniqueness of the limit) Suppose f : D R with x 0 an accumulation point of D. Assume L 1 and L 2 are limits of f at x 0. Prove that L 1 = L Problem 7 in Chapter 2 of Gaughan s book: Define f : (0, 1) R by f(x) = x cos ( 1 x). Does f have a limit at 0? Justify (prove). [Hint: Observe that the cosine function is bounded and hence what can you say about the size of f(x) as x approaches 0?] 5. Problem 9 in Chapter 2 of Gaughan s book: Define f : ( 1, 1) R by f(x) = x + 1 x 2 1. Does f have a limit at 1? Justify (prove). 6. Problem 11 in Chapter 2 of Gaughan s book: Prove the Squeeze Theorem for Limits of Functions: Suppose f, g and h : D R where x 0 is an accumulation point of D, and f(x) g(x) h(x) for all x D, and f and h have limits at x 0 with lim f(x) = x x 0 lim h(x). Prove that g has a limit at x 0 and x x 0 lim x x 0 f(x) = lim x x 0 g(x) = lim x x 0 h(x) 7. Problem 12 in Chapter 2 of Gaughan s book: Suppose f : D R has a limit at x 0. Prove that f : D R has a limit at x 0 and that lim f(x) = x x 0 lim f(x) x x 0. Additional Problems (do not turn in): 1. Problem 6 in Chapter 2 of Gaughan s book. Compare this with Problem 7 in Chapter 2. (#) 2. Problem 10 in Chapter 2 of Gaughan s book. Follow the given hint and use the result of Example 1.10 in Gaughan s book. (#, $) 7

8 3. Problems 1 and 8 in Chapter 2 of Gaughan s book: for those of you who needs additional practice. (#) Homework 7 Due Date: Tuesday, 07/12/ Problem 8 in Chapter 2 of Gaughan s book: Define f : (0, 1) R by Prove that f has a limit at 1 in two ways: f(x) = x3 x 2 + x 1. x 1 (a) Directly using the definition (with ɛ s and δ s). (b) Using the limit theorems in section Problem 14 in Chapter 2 of Gaughan s book: Define f : R R as follows: f(x) = { 8x, if x is a rational number. 2x 2 + 8, if x is an irrational number. Use sequences (and Theorem 2.1) to guess at which points f has a limit, and then use ɛ s and δ s to justify your conclusions. (Also, at points where f does not have a limit, use sequences (and Theorem 2.1) to prove that the limit does not exist at those points. Hint: One approach is to use problems 45, 46 and 39 in Chapter 1 and the main theorem on subsequences (Theorem 1.14) along with the basic limit theorems in section 1.3 to show that f does not have a limit at those points). 3. Project 2.2 in Gaughan s book. Please turn in only parts 1 and 5. But doing the other parts will be essential in understanding the problem better. Let me restate the statements of parts 1 and 5 below. (a) Suppose f : A R and g : B R are such that A B =, x 0 is an accumulation point of A and also an accumulation point of B. Define h : A B R by h(x) = f(x) if x A and h(x) = g(x) if x B. Prove that h has a limit at x 0 if and only if f and g each have a limit at x 0 and lim f(x) = lim g(x). x x 0 x x 0 (b) State and prove a theorem that relates the left-hand limit and right-hand limit of a function and the limit of a function. 4. Problem 17 in Chapter 2 of Gaughan s book: Define f : R R as follows: { x [x], if [x] is even. f(x) = x [x + 1], if [x] is odd. 8

9 Determine those points where f has a limit, and justify your conclusions. [You might use the notation x instead, if you are more comfortable with that.] [Hints: A useful observation is that [x + 1] = [x] + 1. Use this observation to help you with sketching the graph of the function easily (also see additional problem 1). And you might very well use the result of the previous problem on left-hand and right-hand limits in order to prove the existence of limits at certain points.] 5. Problem 19 in Chapter 2 of Gaughan s book: Define f : (0, 1) R by f(x) = Prove that f has a limit at 0 and find it. 9 x 3. x 6. Problem 20 in Chapter 2 of Gaughan s book: Prove Theorem Problem 22 in Chapter 2 of Gaughan s book: Show by example that even though f and g fail to have limits at x 0, it is possible for f +g to have a limit at x 0. Give similar examples for fg and f g. Additional Problems (do not turn in): 1. Problem 13 in Chapter 2 of Gaughan s book: Define f : R R by f(x) = x x (where the function g : R Z defined by g(x) = x is the floor function, that is, x is the greatest integer smaller than or equal to x. Gaughan uses the notation [x] instead - please see Example 2.6. You may use either notation, whichever you feel comfortable with. Please do look up floor and ceiling functions so you understand them clearly and do not confuse each other). Determine those points at which f has a limit, and justify (prove) your conclusions. [Hint: Please take your time to carefully sketch the graph of f. Once you see the picture, the result seems more obvious.] (#) 2. Problem 15 in Chapter 2 of Gaughan s book: Let f : D R with x 0 as an accumulation point of D. Prove that f has a limit at x 0 if for each ɛ > 0, there is a neighborhood Q of x 0 such that for any x, y Q D with x x 0, y x 0, we have f(x) f(y) < ɛ. [Hint: Use Theorem 2.2]( ) [Note that the result of this problem is very useful. The contrapositive of the statement is useful in determining if a function does not have a limit at a point. Also does the converse of the statement hold? If so, prove it! If not, give a counterexample. Hint: (Triange Inequality + (ɛ/2))-argument! ] 3. Problems 16 and 18 in Chapter 2 of Gaughan s book. (#) These types of problems are basic. You would have seen these problems in your Freshmanlevel Calculus classes and you would have known the basic techniques and tricks to find the limits (Recall how you would handle the indeterminate forms 0 0,, 0, etc. by factoring, rationalizing the radical, dividing by the highest power of x in the denominator, etc.). In this class, you should not only know these techniques and tricks but also know how and why they work, that is, you should be able to prove that the limits are what you claim they are based on the theorems that are proved so far. NOTE carefully that you are NOT allowed to use the L Hospital s Rule until we cover it in class formally (section 4.4). 9

10 Homework 8 Due Date: Thursday, 07/14/ Problem 24 in Chapter 2 of Gaughan s book: Let f : [a, b] R be monotone. Prove that f has a limit both at a and at b. [Hint: Let U(a) = inf{f(y) : y > a}. Prove that lim f(x) = U(a). And similarly x a define L(b) to be the supremum of an analogous set (see Lemma 2.7) and show that f(x) = L(b)]. lim x b 2. Problem 2 in Chapter 3 of Gaughan s book: Define f : [ 4, 0] R by f(x) = 2x2 18 x + 3 for x 3 and f( 3) = 12. Show that f is continuous at Problem 3 in Chapter 3 of Gaughan s book: Use Theorem 3.1 to prove that { n e n+1 } n=1 is convergent, and find its limit. You may assume that the function f(x) = e x is continuous on R. 4. Problem 4 in Chapter 3 of Gaughan s book: If x 0 E, x 0 is NOT an accumulation point of E, and f : E R, prove that, for every sequence {x n } n=1 converging to x 0 with x n E for all n, the sequence {f(x n )} n=1 converges to f(x 0). [Remark: If you are someone who still has some trouble understanding accumulation points, I hope this problem can help you understand better.] 5. Problem 5 in Chapter 3 of Gaughan s book: Define f : (0, 1) R by f(x) = 1 x + 1 x x. Can one define f(0) to make f continuous at 0? Explain. 6. Problem 8 in Chapter 3 of Gaughan s book: Suppose f : (a, b) R is continuous and f(r) = 0 for each rational number r (a, b). Prove that f(x) = 0 for all x (a, b). 7. Problem 10 in Chapter 3 of Gaughan s book: Suppose f : E R is continuous at x 0, and x 0 F E. Define g : F R by g(x) = f(x) for all x F. (The function g is often called as the restriction of f to F ). Prove that g is continuous at x 0. Then show by example that the continuity of g at x 0 need not imply the continuity of f at x Problem 11 in Chapter 3 of Gaughan s book: Define f : R R by f(x) = 8x if x is rational and f(x) = 2x if x is irrational. Prove from the definition of continuity that f is continuous at 2 and discontinuous at 1. Additional Problems (do not turn in): 1. Give an example of a monotone function defined on a closed interval [a, b] that fails to have a limit at a countably infinite number of accumulation points of its domain. [Hint: The easiest example I can think of is a function defined on [0, 1] that looks like an infinite staircase and that has jumps at all the points of the form for some n natural number n. Make this idea precise and define an appropriate function carefully. (Feel free to modify my idea to create other such functions that appeal to you better).] 10

11 2. All the remaining problems given under section 3.1 in Gaughan s book. They are all important for your general understanding of continuous functions. Homework 9 Due Date: Tuesday, 07/19/ Problem 13 in Chapter 3 of Gaughan s book: Let f : D R be continuous at x 0 D. Prove that there is M > 0 and a neighborhood Q of x such that f(x) M for all x Q D. [Comment/Hint: The problem states that a function continuous at a point is locally bounded (that is, bounded in a neighborhood of x 0 ). How does it relate to Theorem 2.3?] 2. Problem 14 in Chapter 3 of Gaughan s book: If f : D R is continuous at x 0 D, prove that the funcion f : D R defined by f (x) = f(x) is continuous at x 0. [Comments/Hints: Compare with problem 10 in Chapter 1 and problem 12 in Chapter 2 of Gaughan s book. Try to understand the various ways of proving this result analogous to the various ways of proving Theorem 3.2 that we discussed in class. You will eventually realize that the reverse triangle inequality (part (iv) in Theorem 0.25) is the mother of all the various different arguments. Yet another way of proving the result is to first prove that g : R R defined by g(x) = x is continuous (using any of the several methods). And then observe that f is simply the composite function, g f. There is also another interesting way of proving the fact that g(x) = x is continuous. First prove the following lemma. Lemma 0.1. Suppose f, g : R R are continuous functions on all of R such that f(0) = g(0), then the new function, h : R R defined by h(x) = f(x) if x < 0 and h(x) = g(x) if x 0 is continuous on (all of) R. Then from the lemma, deduce that g : R R defined by g(x) = x is continuous. For the homework, just turn in ONE proof.] 3. Problem 15 in Chapter 3 of Gaughan s book: Suppose f, g : D R are both continuous on D. Define h : D R by h(x) = max{f(x), g(x)}. Show that h is continuous on D. [Hint: First prove that for any two real numbers a, b one has max{a, b} = a + b 2 + a b 2 (Intuitively, the maximum of two numbers is the average value plus half the absolute value of their difference). Then simply use the previous problem along with the other theorems that you learned in this section.] [Comment: (DO NOT TURN IN) Can you prove a similar result for min{f, g} by an analogous argument? ] 11

12 4. Problem 16 in Chapter 3 of Gaughan s book: NOTE carefully that there is a typo in Gaughan s statement. should read as follows: The correct statement Assume the continuity of f(x) = e x and g(x) = ln x. Define h(x) = x x by x x = e x ln x. Show that h is continuous for x > Problem 21 in Chapter 3 of Gaughan s book: Define f : [3.4, 5] R by f(x) = 2 x 3. Show that f is uniformly continuous on [3.4, 5] without using Theorem that is, use the methods of Example Show that the same function as in the previous problem is NOT uniformly continuous on its domain if its domain is changed to (3, 5]. The argument is similar to that in Example Problem 19 in Chapter 3 of Gaughan s book: Let f, g : D R be uniformly continuous. Prove that f + g : D R is uniformly continuous. What can be said about fg? Justify. 8. Problem 26 in Chapter 3 of Gaughan s book: Let E R. Prove that E is closed if, for every x 0 such that there is a sequence {x n } n=1 of points in E converging to x 0, it is true that x 0 E. In other words, prove that E is closed if it contains all limits of sequences of members of E. 9. Problem 28 in Chapter 3 of Gaughan s book: Let D R and let D be the set of all accumulation points of D. Prove that D = D D is closed and that if F is any closed set that contains D, then D F. D is called the closure of D. 10. Problem 32 in Chapter 3 of Gaughan s book: If D R, then x D is said to be an interior point of D iff there is a neighbourhood Q of x such that Q D. Define D to be the set of interior points of D. Prove that D is open and if S is any open set contained in D, then S D. D is called the interior of D. [Comments on problems 9 and 10: Observe that the conclusions of problems 9 and 10 can be rephrased as (a) The closure of a set is the smallest closet set containing it. (b) The interior of a set is the largest open set contained in it. We deduce that a set is closed if and only if its closure is itself and similarly, a set is open if and only if its interior is itself.] Additional Problems (do not turn in): 1. Problem 17 in Chapter 3 of Gaughan s book: Suppose f : D R with f(x) 0 for all x D. Show that, if f is continuous at x 0, then f is continuous at x 0. [Hints: (a) See problem 6 in Chapter 3 of Gaughan s book and use Theorem 3.4. (b) See problem 28 in Chapter 1 and read the first paragraph in page 76 of Gaughan s book.] 2. Problem 22 in Chapter 3 of Gaughan s book: Define f : (2, 7) R by f(x) = x 3 x + 1. Show that f is uniformly continuous on (2, 7) without using Theorem 3.8, that is, use the methods of Example

13 3. Suppose that f : D R is Lipschitz on D. That is, there is some K > 0 such that if x, y D, then f(x) f(y) K x y. (Please see page 70 in Gaughan s book). Prove that f is uniformly continuous (and hence continuous) on D. 4. Problem 20 in Chapter 3 of Gaughan s book: Let f : A B and g : B C be uniformly continuous. What can be said about g f : A C? Justify. [Hint: Theorem 3.4 is analogous.] 5. Prove that a uniformly continuous function preserves Cauchy sequences. That is, if f : D R is uniformly continuous and if {a n } n=1 is a Cauchy sequence, then the sequence {f(a n )} n=1 is Cauchy. [Remark 1: See how this result is used in the proof of Theorem 3.5. Remark 2: See page 70 and our discussion on Lipschitz functions where we proved that a Lipschitz function preserves Cauchy sequences in a similar (but much simpler) way.] 6. All the other problems involving open and closed sets are important for your understanding of these abstract concepts. For now, think about problems 27 and 29. You can ignore the problems 30 and 31 for the time being, but come back to them during the next homework assignment. 7. This is a follow up to problems 9 and 10. This is not super important for the class, but it helps a lot in understanding accumulation points, isolated points, interior points, open sets, closed sets, interior and closure. Let D R. Let D C denote the complement of D in R, that is, D C = R \ D. Let b R. We say that b is a boundary point of D iff for every neighborhood U of b, U D and U D C. That is, b is a boundary point if and only if every neighborhood of b contains points inside the set, D and points outside the set, D. Let bd D denote the set of all boundary points of D. It is called as the boundary of D. Also, let isl D = D \ D, that is, isl D is the set of all isolated points of D. (We say that x is an isolated point of D if x D and there is a neighborhood, U of x such that D U = {x}, and hence x is an isolated point of D if and only if x D \ D.) Prove the following: (a) bd D = bd D C. (b) bd D = D D C (c) bd D = D \ D (d) isl D bd D. (e) D D. (f) D D isl D. (g) D D bd D. (h) D = D isl D. (i) D = D bd D. (j) bd D \ isl D = D \ D = (bd D) D 13

14 (k) Let bdacc D denote the set in the previous part, that is, bdacc D = (bd D) D. Then D = D (bdacc D) (isl D). And observe (prove) that the three sets D, bdacc D and isl D are mutually disjoint. Hence the three sets partition the closure, D. Also observe that D = D (bdacc D) and (bd D) = (isl D) (bdacc D), as seen in the previous part. Please also read the comments towards the end of this problem to understand the partitioning above in a more intuitive way. (l) D is closed if and only if (bd D) D. (m) D is open if and only if D bd D =. Please also read the Wikipedia article on Boundary (topology). There is a nice conceptual Venn diagram given in the article. Essentially, you can think of any subset of the real numbers being made of three kinds of points - isolated points, interior points and the third kind of points which are both boundary points and accumulation points. An interior point is always an accumulation point (as there is too much clustering). An isolated point is always a boundary point (as there is too much scattering - too sparse - that is it is the lone point of the set contained in any small neighborhood of it). Thus, an interior point and an isolated point are in two opposite ends. And both these kinds of points belong to the set. In the middle we have accumulation points which are also boundary points. These are the points which may or may not belong to the set. There is clustering near these points because every neighborhood has infinitely many points of the set; but there isn t TOO much clustering, that is, there is some sparse area, because every neighborhood contains points that are NOT in the set. When all of these points in the middle belong to the set, then the set is closed. Thus, the closure of the set is the union of these three kinds of points. (Please see part (k) above.) Homework 10 Due Date: Thursday, 07/21/ Problem 34 in Chapter 3 of Gaughan s book: Find an open cover of (1, 2) with no finite subcover. 2. Problem 36 in Chapter 3 of Gaughan s book: If E 1, E 2,..., E n are compact, prove that E = n E i is compact. That is, a finite union of compact sets is compact. i=1 Comment: Please try to prove this directly using the open cover definition of compactness and without using the Heine-Borel Theorem. 3. Problem 30 in Chapter 3 of Gaughan s book: Suppose f : R R is continuous and r 0 R. Prove that {x : f(x) r 0 } is an open set. Comment/Hint: Observe how Lemma 3.3 manifests its majesty through this seemingly bizarre looking result! 14

15 4. Problem 23 in Chapter 3 of Gaughan s book: A function f : R R is periodic if and only if there is a real number h 0 such that f(x + h) = f(x) for all x R. Prove that if f : R R is periodic and continuous, then f is uniformly continuous. Hint: Break the domain and use Theorem 3.8. Then give an ɛ δ proof. 5. Problem 37 in Chapter 3 of Gaughan s book: Let f : [a, b] R have a limit at each x [a, b]. Prove that f is bounded. Hint: (Warning: Please do not read the hint until you really need help. Struggling by yourself helps you mature mathematically). Use the (obvious) theorem in Chapter 2 that is relevant and then use the fact that [a, b] is compact. Construct an open cover and extract a finite subcover. 6. Problem 38 in Chapter 3 of Gaughan s book: Suppose f : D R is continuous with D compact. Prove that {x : 0 f(x) 1} is compact. Hint: Use the Heine-Borel Theorem. 7. Problem 39 in Chapter 3 of Gaughan s book: Suppose that f : R R is continuous and has the property that for each ɛ > 0, there is some M > 0 such that if x M, then f(x) < ɛ. (We say that f vanishes at infinity if f has this property. Think of examples!). Show that f is uniformly continuous. Hint: (Warning: Again, please do not read the hint until you really need help. Struggling by yourself helps you mature mathematically). [ M, M] is compact! Use Theorem 3.8 along with an ɛ/3-argument. 8. Problem 41 in Chapter 3 of Gaughan s book: Find an interval of length 1 that contains a root of the equation xe x = 1. Comment: You will need a calculator. 9. Problem 44 in Chapter 3 of Gaughan s book: Suppose that f : [a, b] [a, b] is continuous. Prove that there is at least one fixed point in [a, b] - that is, x such that f(x) = x. Hint: Consider g(x) = f(x) x. Apply the Intermediate Value Theorem. Additional Problems (do not turn in): 1. Problem 31 in Chapter 3 of Gaughan s book: Suppose that f : [a, b] R and g : [a, b] R are both continuous. Let T = {x : f(x) = g(x)}. Prove that T is closed.(#, ) Comment: Observe that f g is continuous and T = {x : (f g)(x) = 0}. It is not required that you use the observation for the proof, but the observation gives a specific instance of a more general result which you will learn in a topology class if you plan to take one in future. 2. Problem 33 in Chapter 3 of Gaughan s book: Find an open cover of {x : x > 0} with no finite subcover. (#) 3. Problem 35 in Chapter 3 of Gaughan s book: Let E be compact and nonempty. Prove that E is bounded and that sup E and inf E both belong to E. ( ) Comment: This is an important result that is very easy to prove. 4. Problem 25 in Chapter 3 of Gaughan s book: Give an example of sets A and B and a continuous function f : A B R such that f is uniformly continuous on A and uniformly continuous on B, but not uniformly continuous on A B. (#) Comment/Hint: This is a harder problem due to it being counterintuitive. But, it gives a 15

16 good understanding of the difference between continuity and uniform continuity. Please do not spend too much time on this. The hint is to consider a set containing only isolated points along with a single accumulation point that does not belong to the set. Then break this set into two pieces A and B. Define the function f to be a constant on A and a different constant on B. Then show that f has the required properties. (Recall problems 2 and 3 in Worksheet 5). 5. Problem 40 in Chapter 3 of Gaughan s book: Give an example of a function f : R R that is continuous and bounded but not uniformly continuous. (#) Hint: Think of a bounded function whose derivative (slope of the tangent line) at certain points tend to infinity as x. 6. Problem 42 in Chapter 3 of Gaughan s book: This is very similar to problem 41. (#) 7. Problem 45 in Chapter 3 of Gaughan s book: If f : [a, b] R is 1 1 (injective) and has the intermediate-value property that is, if y is between f(u) and f(v), then there is an x between u and v such that f(x) = y show that f is continuous. Hint: First show that f is monotone and then recall Lemma 2.7 and Theorem 2.8. Observe that f is continuous at every point in (a, b), (not the closed interval [a, b]), where the limit exists (why?). Then show that the limit exists at every point in (a, b). Finally show that f is also continuous at the end points a and b. Homework 11 Due Date: Tuesday, 07/26/ Prove the converse of Problem 26 in Chapter 3 of Gaughan s book: That is, prove the following statement: Let E R. If E is closed, then for every x 0 R such that there is a sequence {x n } n=1 of points of E converging to x 0, it is true that x 0 E. Comment: Combining this with the Problem 26, we have the result: Let E R. Then E is closed if and only if for every x 0 R such that there is a sequence {x n } n=1 of points of E converging to x 0, it is true that x 0 E. That is, E is closed if and only if E contains all limits of sequences of members of E. Note that we can also rephrase the statement as follows: E is closed if and only if for every convergent sequence {x n } n=1 with x n E for all n and converging to a limit x 0 R, we must have that x 0 E. 16

17 Comment: Observe how the above result is useful in / related to proving the statements given in the next problem and its comments. 2. Prove the converse of the lemma given in page 97. That is, prove that if for every sequence {x n } n=1 such that x n E for all n there exists some x 0 E and a subsequence {x nk } k=1 that converges to x 0, then E is compact. Hint:A proof by contradiction is an easy approach. You may freely use the Heine-Borel Theorem. (You could to try to prove it WITHOUT USING the Heine-Borel Theorem (or the argument in the Heine-Borel theorem that proves closed+bounded implies compact ) and prove it DIRECTLY USING THE OPEN COVER DEFINITION of compactness. BUT this is VERY challenging!!! I advise you NOT to spend too much time on this, at least until the end of the semester.) COMMENTS/PROJECT: I strongly urge you to read the following comments even if you do not plan on taking more classes in analysis or topology in future. I believe such theorems on equivalent statements helps in building your intuition and familiarizing yourselves with the various ways of thinking about compact sets so that you can immediately recognize when certain sets are compact and when they are not. Definition 0.2. Let F R. We say that F is sequentially compact if for every sequence {x n } n=1 such that x n F for all n there exists some x 0 F and a subsequence {x nk } k=1 that converges to x 0. (Informally, in short, we say that F is sequentially compact if and only if every sequence in F has a convergent subsequence that converges to some point in F. (Note that the last two words in the previous sentence are very important).) Thus the lemma along with the converse proven in this problem implies the following theorem. Theorem 0.3. Let E R. Then E is compact if and only if E is sequentially compact. The above theorem along with the Heine-Borel Theorem can be stated together as the following theorem. Theorem 0.4. Let E R. Then the following are equivalent: (a) E is compact. (b) E is sequentially compact. (c) E is closed and bounded. 17

18 There is yet another equivalent statement. If you are really interested in this stuff, I strongly urge you to do the following problem. Additional Problem: (DO NOT TURN IN!) Let E R. Prove that E is compact if and only if E is complete and totally bounded (These terms are defined below). You may use the Heine-Borel Theorem or the theorems stated above in order to prove this. A useful exercise is to (a) First prove the statement by showing that complete and totally bounded if and only if closed and bounded (Hint: see observations/exercises given below: show that closed if and only if complete and that bounded if and only if totally bounded ) (b) Next prove the same statement by showing that complete and totally bounded if and only if sequentially compact (Hint: See observations/exercises given below: show that a subset of a totally bounded set is totally bounded and then use it to show that totally bounded if and only if Cauchy-precompact which is challenging but the main idea is to argue similarly as in the proof of the Bolzano-Weierstrass Theorem. If you find the above exercise hard, just assume for the moment that totally bounded if and only if Cauchy-precompact is true and use it to prove complete and totally bounded if and only if sequentially compact and see how easy the proof becomes with this assumption.) (But proving the statement DIRECTLY USING THE OPEN COVER DEFINITION OF COMPACTNESS is VERY challenging!!! Again, I advise you NOT to spend too much time on this at least until the end of the semester.) Definition 0.5. Let E R. We say that E is complete if and only if every Cauchy sequence in E converges to a point in E, that is, for every Cauchy sequence {x n } n=1 with x n E for all n, there exists x 0 in E such that {x n } n=1 converges to x 0. Observations/Exercices: (a) Observe that the phrase the limit x 0 is in the set E is very crucial for the definition. (b) Give examples of subsets of R that are not complete. (c) Give examples of subsets of R that are bounded but not complete. (d) Read the comment below Problem 1 in this homework assignment and read the definition of a complete subset of R again. Observe the similarities and differences. Can you say that a complete subset of R is always closed? What about the converse? Justify! Definition 0.6. Let E R. We say that E is totally bounded if for every ɛ > 0, there are a finite number of points x 1, x 2,..., x n R (for some n N) such that n E (x i ɛ, x i + ɛ). i=1 18

19 Observations/Exercises: (a) Prove that a subset of a totally bounded set is totally bounded. (b) Is a totally bounded subset of R always bounded? What about the converse? Justify! (c) We say that a subset F of R is Cauchy-precompact if every sequence in F has a Cauchy subsequence. Prove that a subset E of R is Cauchy-precompact if and only if it is totally bounded. (This is a challenging exercise!! Use part (a)). Finally, once you proved the additional problem, we observe that we can put this together with the previous theorems into one grand theorem as follows: Theorem 0.7. Let E R. Then the following are equivalent: (a) E is compact. (b) E is complete and totally bounded. (c) E is sequentially compact. (d) E is closed and bounded. If you are going to ask, if all of these statements are equivalent for subsets of R, then why did we need all those crazy definitions, my answer is that hopefully I have motivated you to study more abstract topological spaces for whose subsets the above statements are not always equivalent. (Of course, in abstract topological spaces, the definitions of the terms are analogous but different from the definitions of the terms for subsets of R). 3. Problem 1 in Chapter 4 of Gaughan s book: Let (x 0, y 0 ) be an arbitrary point on the graph of the function f(x) = x 2. For x 0 0, find the equation of the line tangent to the graph of f at that point by finding a line that intersects the curve at exactly one point. Do NOT use the derivative to find this line. Homework 12 Due Date: Thursday, 07/28/ In this problem you will explore whether the unions and intersections of open/closed sets are open/closed. PLEASE TURN IN ONLY PARTS (a), (b), (e) and (f). (Parts (c) and (d) are VERY useful exercises too. Please make sure you think about them and see how your arguments in parts (a) and (b) generalize. But please do not turn them in). (a) Let A and B be open subsets of R. Prove that A B and A B are open. (b) Let A and B be closed subsets of R. Prove that A B and A B are closed. [Hint: You may use the De Morgan s Laws along with Theorem 3.6 to prove part (b) once you have proven part (a), but a direct proof is much more illuminating about what s going on]. 19

20 (c) Prove that an arbitrary union of open sets is open. But only a finite intersection of open sets is open. That is, prove the following: i. If {A λ } λ Λ is a family of open subsets of R, then λ Λ A λ is open (no matter how HUGE the index set Λ is it could be uncountable!). ii. If A 1, A 2,..., A n are open subsets of R (for some n N), then n A i is open. [Hint: You may use induction. You proved the base case in part (a). But a direct proof (without using induction) analogous to your proof in part (a) is very illustrative of the characteristic feature of the real line. Hence I strongly encourage you to write down a direct proof]. (d) Prove that an arbitrary intersection of closed sets is closed. But only a finite union of closed sets is closed. That is, prove the following: i. If {A λ } λ Λ is a family of closed subsets of R, then λ Λ i=1 A λ is closed (no matter how HUGE the index set Λ is it could be uncountable!). ii. If A 1, A 2,..., A n are closed subsets of R (for some n N), then n A i is closed. [Hint: Again, you may use induction. You proved the base case in part (b). But again, a direct proof (without using induction) analogous to your proof in part (b) is very illustrative of the characteristic feature of the real line. Hence I strongly encourage you to write down a direct proof]. [Hint: Observe that you might also use the De Morgan s Laws along with Theorem 3.6 to prove this part once you have proved part (c)]. (e) Give an example to show that if {A n : n N} is a family of open subsets of R then A i need not be open. i=1 (f) Give an example to show that if {A n : n N} is a family of closed subsets of R then A i need not be closed. i=1 2. Problem 2 in Chapter 4 of Gaughan s book: Prove that the definition of the derivative and the alternate definition of the derivative are equivalent, that is, prove that and in the case when the limits exist. lim x x 0 T (x) exists if and only if lim t 0 Q(t) exists, lim T (x) = lim Q(t) x x 0 t 0 3. Problem 3 in Chapter 4 of Gaughan s book: Use the definition to find the derivative of f(x) = x, for x > 0. Is f differentiable at 0? 4. Problem 5 in Chapter 4 of Gaughan s book: Define h(x) = x 3 sin 1 x for x 0 and h(0) = 0. Show that h is differentiable everywhere and that h is continuous everywhere but fails to have a derivative at one point. You may use the rules for differentiating products, sums and quotients of elementary functions that you learned in calculus. i=1 20

21 5. Problem 7 in Chapter 4 of Gaughan s book: A function f : (a, b) R satisfies a Lipschitz condition at x (a, b) iff there is M > 0 and ɛ > 0 such that x y < ɛ and y (a, b) imply that f(x) f(y) M x y. Give an example of a function that fails to satisfy a Lipschitz condition at a point of continuity. If f is differentiable at x, prove that f satisfies a Lipschitz condition at x. [Hint: Recall our discussion on (globally) Lipschitz functions in Chapter 2 and Chapter 3. Recall the the Lipschitz condition essentially says that the slopes of the secant lines are bounded. What the problem states is that if f is differentiable at x, then it must be locally Lipschitz at x. A function fails to be locally Lipschitz at a point x if the slope of the tangent line is infinite at that point, that is, there is a vertical tangent line at the point.] 6. Problem 9 in Chapter 4 of Gaughan s book: Suppose f : (a, b) R is continuous on (a, b) and differentiable at x 0 (a, b). Define g(x) = f(x) f(x 0) x x 0 for x (a, b) \ {x 0 }, g(x 0 ) = f (x 0 ). Prove that g is continuous on (a, b). 7. Problem 11 in Chapter 4 of Gaughan s book: Prove that f : (0, 1) R defined by f(x) = 2x 2 3x + 6 is differentiable on (0, 1) and compute the derivative. 8. Problem 14 in Chapter 4 of Gaughan s book: Suppose f : R R is differentiable and define g(x) = x 2 f(x 3 ). Show that g is differentiable and compute g. Additional Problems (do not turn in): 1. Problem 4 in Chapter 4 of Gaughan s book: Use the definition to find the derivative of g(x) = x Problem 6 in Chapter 4 of Gaughan s book: Suppose f : (a, b) R is differentiable at x (a, b). Prove that f(x + h) f(x h) lim h 0 2h exists and is equal to f (x). Give an example of a function where this limit exists, but the function is not differentiable. Homework 13 Due Date: Tuesday, 08/02/ Problem 17 in Chapter 4 of Gaughan s book: Define f : R R by f(x) = 1 1+x 2. Prove that f has a maximum value and find the point at which the maximum occurs. 21

22 2. Problem 19 in Chapter 4 of Gaughan s book: Show that cos x = x 3 + x 2 + 4x has exaclty one root in [0, π 2 ]. [Hint: First show that there is at least one root and then show that there can not be more than one root.] 3. Problem 21 in Chapter 4 of Gaughan s book: Let f : [0, 1] R and g : [0, 1] R be differentiable with f(0) = g(0) and f (x) > g (x) for all x in [0, 1]. Prove that f(x) > g(x) for all x in (0, 1]. 4. Problem 25 in Chapter 4 of Gaughan s book: Suppose that f : (a, b) R is differentiable and f (x) M for all x (a, b). Prove that f is uniformly continuous on (a, b). Give an example of a function f : (0, 1) R that is differentiable and uniformly continuous on (0, 1) but such that f is unbounded. [Comment/Hint: Observe how our intuition for uniform continuity is challenged in this problem. On one hand, it is intuitive that if the magnitudes of the slopes of the tangent lines are bounded, ( then the function is uniformly continuous since we may simply choose ɛ ) our δ to be and that works for all x in the domain of f. We may simply follow M this intuition rigorized with the help of the Mean Value Theorem to prove the first part. On the other hand, the second part asks you to prove that converse is not true in general. That is, you are asked to prove that it is NOT NECESSARY that THE DERIVATIVE IS BOUNDED in order for the function to be uniformly continuous. Note that your example function CAN NOT BE UNBOUNDED, due to Theorem 3.9 and similarly, your example function MUST HAVE A LIMIT at ALL accumulation points of the domain (0, 1), due to Theorem 3.5. This is why it might be a bit counterintuitive and challenging. But it is not hard to find such an example. The problem thus exhibits the need for such precision and rigour in mathematics so that we are careful enough not to misunderstand things by relying solely on our intuition.] 5. Problem 26 in Chapter 4 of Gaughan s book: Suppose f is differentiable on (a, b), except possibly at x 0 (a, b) and is continuous on [a, b]; assume that lim f (x) exists. Prove x x 0 that f is differentiable at x 0 and f is continuous at x 0. [Hint: Use Theorem 4.1, the Mean Value Theorem, Theorem 2.1 and Theorem 3.1.] 6. Problem 28 in Chapter 4 of Gaughan s book: Prove that the function f(x) = 2x 3 + 3x 2 36x + 5 is 1 1 (injective) on the interval [ 1, 1]. Is f increasing or decreasing on that interval? 7. Problem 31 in Chapter 4 of Gaughan s book: If f : [a, b] R is differentiable at c, where a < c < b and f (c) > 0, prove that there is x such that c < x < b and f(x) > f(c). 8. Problem 33 in Chapter 4 of Gaughan s book: Use L Hospital s Rule to find the limit ( x 2 ) sin x lim x 0 sin x x cos x [Hint/Warning: Do not keep applying the L Hospital s Rule repeatedly assuming that you will eventually find the limit after a finite number of steps. This need not happen. On the other hand, you can not conclude that the limit does not exist just because L Hospital s Rule does not lead you to a limit in a finite number of steps.] 22