Quiz 8 Possible Questions
29.15
Let r be a nonzero rational number
m
n
where n is a positive integer, m is any nonzero
integer, and m and n have no common factors.
h ( x )=x r where dom(h) = [0,) if n is even and m > 0, dom(h) = (0,) if n is even
Let
and
Real Analysis 1 : Final Prep Definitions
Numbers
Natural number - cfw_1,2,3, all positive integers has a successor, + 1
N1. 1 belongs to
N2. If belongs to , then its successor + 1 belongs to
N3. 1 is not the successor of any element in .
Rational Number
Real Analysis 1 : Midterm 2 Notes
Sequences (sn)
Bounded : For cfw_sn : n , M s.t. |sn | M n
Decreasing : sn sn+1 n , sn sm whenever n < m
Increasing : sn sn+1 n , sn sm whenever n < m
Monotonic or Monotone: any sequence that is increasing or decreasing
Math 131 B, Lecture 1
Real Analysis
Sample Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward corre
Math 131 B, Lecture 1
Real Analysis
Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward correct proo
Math 131 B, Lecture 1
Real Analysis
Sample Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward corre
Math 131 B, Lecture 1
Real Analysis
Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward correct proo
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Math 131A: General Course Outline
Catalog Description
131A. Analysis. (4) Lecture, three hours; discussion, one hour. Requisites: co
Math 131B-2
Homework 7
Due: December 2, 2016
This assignment is due on December 2, 2016 in class . Provide complete well-written solutions to the following
exercises.
Exercise 1. In this exercise, you will give another proof of Theorem 42 from the notes.
Math 131B-2
Homework 1
Due: October, 4th, 2016
This assignment is due on October 4th, 2016 in the discussion section. Provide complete well-written solutions to
the following exercises.
Exercise 1. Let H be a vector space over R and let h, i be an inner-p
Math 131B-2
Homework 3
Due: October, 18th, 2016
This assignment is due on October 18th, 2016 in the discussion section. Provide complete well-written solutions to
the following exercises.
Exercise 1. Consider the following definition.
Definition 1. Let (X
Math 131B-2
Homework 2
Due: October, 11th, 2016
This assignment is due on October 11th, 2016 in the discussion section. Provide complete well-written solutions to
the following exercises.
Exercise 1. Let (X, d) be a metric space and Y X. A collection cfw_
Math 131B-2
Homework 5
Due: November 8, 2016
This assignment is due on November 15th, 2016 in the discussion section. Provide complete well-written solutions
to the following exercises.
Exercise 1. Consider the following definition (which is related, but
Math 131B-2
Homework 5
Due: November 8, 2016
This assignment is due on November 8th, 2016 in the discussion section. Provide complete well-written solutions
to the following exercises.
Exercise 1. Use the Weierstrass approximation theorem to prove the fol
Math 131B-2
Homework 4
Due: October, 25th, 2016
This assignment is due on October 25th, 2016 in the discussion section. Provide complete well-written solutions to
the following exercises. In what follows, we assume the notation of the supplementary course
1. (a) Prove that if an 3 a for all n and limnem 3n : s E R, then
s 2 a. (b) Prove that if an S tn for all n and hm,H00 5n 2: s and
1imn_>00 tn 2 1213116511 8 g t. (Hint: Use part (30.)
cfw_:12 #0111?ij [on SIAWDQ ax m& M 5 rawa >O
damn (Hui/vi WISE N W0/
1. State and prove the Archimedean Property.
2. (a) Define: limn sn = s R. (b) Prove that if an sn bn
for all but finitely many n N and limn an = limn bn = s,
then limn sn = s. Hint: there exists N0 N such that an
sn bn for all n > N0 .
3. (a) Define: th
1. (a) Define: f (x) is uniformly continuous on S R. (b) Prove
that if f (x) is uniformly continuous on (, 0] and also uniformly
continuous on [0, +), then f (x) is uniformly coninuous on all of
R.
2. (a) Define limn xn = +. (b) Define limx+ f (x) = L
by
Mathematics 131A/5
Fall, 2016
MIDTERM 1 INFORMATION
The examination will be at 2:00 on Friday, October 21, NOT in the usual
classroom but in
Dodd 175
It is a closed book examination. You will work entirely on the examination
paper, no scratch paper is per
HOMEWORK 3
MATH 131A
Chapter 2, Section 9: 9.4 9.15.
The following problems are due on Friday, October 28:
Chapter 2, Section 9: 9.6, 9.9, 9.10, 9.11, 9.12.
1
Math 131 A, Lecture 1
Real Analysis
Sample Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward corre
Math 131 A, Lecture 1
Real Analysis
Sample Midterm 2
Instructions: You have 50 minutes to complete the exam. There are five problems, worth
a total of fifty points. Write your solutions in the space below the questions. If a question is in
multiple parts,
Math 131A, Lecture 1
Real Analysis
Final Exam
Instructions: You have three hours to complete the exam. There are ten problems, worth a
total of one hundred points. Write your solutions in the space below the questions. If a question
is in multiple parts,
Math 131 A, Lecture 1
Real Analysis
Sample Midterm 1
Instructions: You have 50 minutes to complete the exam. There are five problems, worth a
total of fifty points. You may not use any books or notes. Partial credit will be given for progress
toward corre
Math 131 A, Lecture 1
Real Analysis
Sample Midterm 2
Instructions: You have 50 minutes to complete the exam. There are five problems, worth
a total of fifty points. Write your solutions in the space below the questions. If a question is in
multiple parts,
Math 131A, Lecture 1
Real Analysis
Sample Final Exam
Instructions: You have three hours to complete the exam. There are ten problems, worth a
total of one hundred points. Write your solutions in the space below the questions. If a question
is in multiple
Math 131A, Lecture 1
Real Analysis
Final Exam
Instructions: You have three hours to complete the exam. There are ten problems, worth a
total of one hundred points. Write your solutions in the space below the questions. If a question
is in multiple parts,