Math 401
Final Exam Review
The following problems are representative of the problems that will appear on the
Final Exam on Tuesday the 12th. Justify your answers.
For problems 5 and 6, if you construct a continuous function, you need not prove
that it is
Math 401
Homework #21 due 12/3
Reading Assignment:
Read section 4.4 (pages 114-119) of Understanding Analysis and make sure the thou
understand the following theorems and denitions:
1. Denition 4.4.1. Given a function f : A R and B A, then f (B ) denotes
Math 401
Homework #19 due 10/01
Reading Assignment:
Read sections 4.1-4.3 (pages 99-113) of Understanding Analysis and make sure the
thou understand the various denitions of the following:
1. For a function f : A R and a limit point c of the domain A, the
Math 401
Homework #18 due 11/19
Reading Assignment:
Read sections 3.5 of Understanding Analysis. We will only cover the parts of this
section that concern dense and nowhere-dense sets. We will not cover G-sigma sets
or Baires Theorem.
Go over your class n
Exercise 2.3.1. (a) Let c > 0 be arbitrary. We must nd an N k such that
n 3 N implies |,/:1:_, 0| < e. Begause (an) > 0, there exists N E N such that
n _>_ N implies |wn 0| = 3, < 2:qung this N, we have (9:7,)2 < 62, which
gives fe? 0| < e for all n 2 N,
Midterm 2, Math 401, October 28, 2015. Name:
Provide complete proofs that follow the rules of logic. In proving a limit equality;r you may
use the standard limits we have proven, as well as the algebraic rules. Of course; you may
also use the squeeze theo
Final, Mat-h 401, Fall 2015. Name:
Provide proofs that follow the rules of logic. In case of disproving a for all statement, pro-
vide a counterexample. All exercises have equalweight. All exercises need full explanation.
Do exactly 5 exercises. Mark the
Midterm 1, Math 401, October 7, 2015. Name: Wm
Provide complete proofs that follow the rules of logic. For instance, in proving an equality
A = B of sets, you need to prove both A Q B and B Q A. In proving an equality a = 13
of real numbers, it often work
Midterm 3, Math 401, November 18, 2015. Name:
Provide complete proofs that follow the rules of logic. In proving a limit
equality you may use the standard limits we have proven, as well as the
algebraic rules. Of course, you may also use the squeeze theor
Exercise 2.3.1. (a) Let c > 0 be arbitrary. We must nd an N k such that
n 3 N implies |,/:1:_, 0| < e. Begause (an) > 0, there exists N E N such that
n _>_ N implies |wn 0| = 3, < 2:qung this N, we have (9:7,)2 < 62, which
gives fe? 0| < e for all n 2 N,
Midterm 1, Math 401, October 7, 2015. Name: Wm
Provide complete proofs that follow the rules of logic. For instance, in proving an equality
A = B of sets, you need to prove both A Q B and B Q A. In proving an equality a = 13
of real numbers, it often work
are twatzr'aareemwwzces 1
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Chapter2 Sequences of Real Numbers
This can also be proved directly. Since lim [9, = 0, given 6 > 0, there exists
nm-DUO
no G N such that 0 S b" < e/(M + 1) for all n 2 no. Thus forn 2 no,
|n,-
Math 401
Homework #6 due 10/10
Reading Assignment:
Read sections 1.1 and 1.2 (pages 1-11) of Understanding Analysis.
Go over your class notes and write out denitions of the following:
1. The set of natural numbers N.
2. The set of integers Z.
3. The set o
Math 401
Homework #4 due 10/03
Reading Assignment (not to be handed in):
Go over your class notes and write out denitions of the following:
1. The distance from a point a to 0 in Rn , i.e. |a|.
2. The distance between two point a and b in Rn , i.e. |a b|.
Math 401
Homework #2 due 9/28
Reading Assignment:
Read section 2.2 (pages 38-43) of Understanding Analysis.
Go over your class notes and write out denitions of the following:
1. A function f : R R is continuous at a point x = A.
General Questions (not to
Math 401
Homework #2 due 9/28
Reading Assignment:
Read over your class notes and go over the following two equivalent denitions:
1. A sequence a converges to a limit A, denoted cfw_ai A, if for each real
i=0
number > 0, there exists some term in the sequ
Math 401
Homework #1 due 9/26
Reading Assignment:
1. Read Chapter 1 (pages 3-5 )of Functional Analysis available through the link
in the email.
General Questions (not to be handed in):
1. Why is it important to understand that a continuous function can be