Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: January 20, 2010
Assignment 1 1. Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n N. We prove this via induction. The base case of n = 1 is trivial to check since the given quantit
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 21, 2010
Assignment 27 1. If f is continuous on [a, b], a < b, show that there exists c [a, b] such that we have f (c)(b a). This result is sometimes called the Mean Value Theorem for Integ
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 24, 2010
Assignment 19 1. If f : R R is continuous, prove that f (E ) f (E ) for every set E R. (E denotes the closure of E as dened on exam 1). Show, by an example, ) can be a proper subse
1.4.7. Finish the following proof for Theorem 1.4.12. Assume B is a countable set. Thus, there exists f : N ! B , which is 1 and onto. Let A B be an innite subset of B . We must show that A 1 is countable. Let n1 = min fn 2 N : f (n) 2 Ag. As a start to a
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 1, 2010
Assignment 14 1. Show that
(a)
n=0
1 = 1, (n + 1)(n + 2) 1 1 = > 0, if > 0, ( + n)( + n + 1) 1 1 =. n(n + 1)(n + 2) 4 To each of these, we apply partial fractions. For (a), we have
1.3.7. Prove that if a is an upper bound for A, and if a is also an element of A, then it must be that a = sup A. Since we are given that a is an upper bound for A, we need only check that part (ii) of Denition 1.3.2, namely that if b is any upper bound f
92.305 Homework 2
Solutions October 2, 2009
Exercise 2.2.1. Verify, using the denition of convergence of a sequence, that the following sequences converge to the proposed limit.
1 (a) lim (6n2 +1) = 0.
Proof: Let > 0 be given. We take any N N strictly gre
92.305 Homework 1
Solutions September 18, 2009
Exercise 1.2.1. (a) Prove that 3 is irrational. Proof: Suppose that 3 is rational. Then we can write 3 = (p/q)2 with p, q Z and q = 0. Without loss of generality, we can arrange this so that p, q share no co
92.305 Homework 4
Solutions October 9, 2009
Exercise 2.4.1. Complete the proof of Theorem 2.4.6 by showing that if the series 2n b2n diverges, then so does bn . Example 2.4.5 may be a useful reference. n=1 n=0 Proof: We will show that if 2n b2n diverges t
92.305 Homework 2
Solutions September 25, 2009
Exercise 1.3.2. (a) Write a formal denition in the style of Denition 1.3.2 for the inmum or greatest lower bound of a set. Answer: A number x R is the inmum or greatest lower bound of a set A R of real number
2.4.2. Dene the sequence (xn ) by x1 = 3 and xn+1 = 1 4 xn
(a) Prove that the sequence (xn ) converges. The rst three terms of the sequence are 3, 1, 1=3. We want to show that this sequence is decreasing (it will be bounded below by 0 because xn 3). So we
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 26, 2010
Assignment 29 1. Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. That is, cfw_fn is a uniformly convergent sequence of functions on a set
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 23, 2010
Assignment 28 1. Use the following argument to prove the Substitution Theorem (Theorem 7.3.8). Dene F (u) = u () f (x) dx for u I , and H (t) = F (t) for t J . Show that H (t) = f
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 19, 2010
Assignment 26 1. Suppose f 0, f is continuous on [a, b], and that x [a, b].
b a f (x) dx
= 0. Prove that f (x) = 0 for all
Suppose not. That is, suppose that f (x) = c > 0 for some
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 12, 2010
Assignment 24 1. Suppose f is dened in a neighborhood of x, and suppose f (x) exists. Show that f (x + h) + f (x h) 2f (x) = f (x). h0 h2 lim Show by an example that the limit may
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: April 5, 2010
Assignment 23 1. A dierentiable function f : I R is said to be uniformly dierentiable on I := [a, b] if for every > 0 there exists > 0 such that if 0 < |x y | < and x, y I then f (x
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 31, 2010
Assignment 22 1. Suppose that f : R R is dierentiable at c and that f (c) = 0. Show that g (x) := |f (x)| is dierentiable at c if and only if f (c) = 0. Since f (c) = 0, we have g
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 29, 2010
Assignment 21 1. Let I R be an interval and let f : I R be increasing on I . If c is not an endpoint of I , show that the jump jf (c) of f at c is given by inf cfw_f (y ) f (x) : x
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 26, 2010
Assignment 20 1. Show that if f and g are uniformly continuous on A R and if they are both bounded on A, then their product f g is uniformly continuous on A. Since f and g are both
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 22, 2010
Assignment 18 1. Let f be a continuous real function on R. Let Z (f ) (the zero set of f ) be the set of all p R at which f (p) = 0. Prove that Z (f ) is closed. The set E = cfw_x
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 19, 2010
Assignment 17 1. Let f : R R be dened by setting f (x) := x if x is rational, and f (x) = 0 if x is irrational. (a) Show that f has a limit at x = 0. (b) Use a sequential argument
Math 521 - Advanced Calculus I
Instructor: J. Metcalfe Due: March 17, 2010
Assignment 16 1. Suppose that none of the numbers a, b, c is a negative integer or zero. Prove that the hypergeometric series ab a(a + 1)b(b + 1) a(a + 1)(a + 2)b(b + 1)(b + 2) + +