Math 401: Homework 1
September 7, 2016
Solutions
Exercise 1.2.5: Use the triangle inequality to establish the following inequalities:
(a) |a b| |a| + |b|;
(b) |a| |b| |a b|.
Solution:
(a) Proof. By the triangle inequality and the fact that |x| = | x| for
Math 401: Homework 2
Due September 12, 2016
Solutions
Exercise 1.4.1: Recall that I stands for the set of irrational numbers.
(a) Show that if a, b Q then ab and a + b Q as well.
(b) Show that if a Q and t I then a + t I and if a , 0 then at I as well.
(c
Math F401: Homework 5 Solutions
October 6, 2016
1. Suppose cfw_n j
j=1 is a sequence of natural numbers such that n j < n j+1 for all j N. Show
that n j j for all j N
2. Show that a subsequence of a convergent sequence converges to the same limit. Be sur
Math F401: Homework 5
Due: October 12, 2016
1. Use the worksheet from class to write up a nice proof of the Alternating Series Test.
Solution:
Let (a n )n be a decreasing sequence of non-negative numbers such that lim a n = 0. We
wish to show that (1)n+1
Math F401: Homework 4
Due: September 29, 2016
1. Abbott 2.2.6
Show that limits, if they exist, must be unique. In other words, assume lim a n = l1 and
lim a n = l2 , and prove l1 = l2 .
Solution:
Let > 0. Since lim a n = l1 , there exists an N1 such that
Math 401: Homework 3
Due September 19, 2016
Solutions
Exercise 1.4.7: Finish the proof of Theorem 1.4.5 by showing that the assumption 2 > 2
contradicts the assumption that = sup A.
Solution:
We first observe that if x > 0 and x2 > 2 then x is an upper bo
Math F401: Homework 9
Due: November 9, 2016
1. Hand in your proof of the Intermediate Value Theorem.
Solution:
Suppose f [a, b] is continuous and f (a) < f (b). Suppose v ( f (a), f (b). Let Av = cfw_x
[a, b] f (x) < v. Observe that a Av and that Av is b
Math F401: Homework 8 Solutions
Due: November 2, 2016
1. Abbott 4.3.7
Solution, part a:
Let c R. Let (x n ) be a sequence of rational numbers coverging to c, and let (y n ) be a
sequence of irrational numbers coverging to c. The existence of these sequenc
Math F401: Introduction to Real Analysis
Fall 2016 Syllabus
Course Description
This course is a rigorous study of the ideas underlying calculus and an introduction to the
real numbers. Rather than the computational focus of your previous calculus classes,