PMath/AMath 331, Fall 2013 - Assignment 1
Handed out on Friday, September 13. Selected questions are due on Friday, September 20
Topics: Denition of limit, Properties of limits, Bounded sequences, Squ
Last name (Print):
UW Student ID Number:
University of Waterloo
Midterm Examination
AMath/PMath 331
(Real Analysis and Applications )
Instructor: Robert Andre
Date: Thursday, November 7, 2013
Term: 11
PMath/AMath 331, Fall 2015 - Assignment 1. Solutions
Posted on Friday, September 18. Due on Friday, September 25 at noon.
Topics: Denition of limit, Properties of limits, Bounded sequences, Squeeze th
PMath/AMath 331, Fall 2013 Solutions for assignment 1
Handed out on Friday, September 13; due on Friday, September 20
Topics: Denition of limit, Properties of limits, Bounded sequences, Squeeze theore
University of
Final Examination
WATERLOO
Fall 2011
Name (Please Print):
UW Student Identication Number:
AMath 331 / PMath 331
Applied Real Analysis
Instructor: Kenneth R. Davidson
Date: Tuesday, Decem
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PMath/AMath 331, Fall 2013 - Solutions for assignment 2
Handed out on Friday, September 20; due on Friday, September 27.
Topics: Upper and lower bounds, Least upper bound principle, Monotone convergen
PMath/AMath 331, Fall 2013 - Assignment 5
Handed out on Friday October 11; due on Friday October 18.
Topics: Open and closed subsets of a normed vector space.
Problem 1. Show by using the denition of
PMath/AMath 331, Fall 2013 - Assignment 4.
Handed out on Friday October 4; due on Friday, October 11.
Topics: Normed vector spaces, convergence and completeness in Rn .
Problem 1
a) Show that if that
PMath/AMath 331, Fall 2013 - Solutions for assignment 5
Handed out on Friday October 11; due on Friday October 18.
Topics: Open and closed subsets of a normed vector space.
Problem 1. Show by using th
PMath/AMath 331, Fall 2013 Solutions for assignment 4
Handed out on Friday October 4; due on Friday, October 11.
Topics: Normed vector spaces, convergence and completeness in Rn .
Problem 1 a) Show th
PMath/AMath 331, Fall 2013 - Assignment 6
Handed out on Friday October 18; due on Friday October 25.
Topics: Compact sets, Heine-Borel theorem, continuous functions on Rn .
Submit questions 1 to 4, 6,
PMath/AMath 331, Fall 2013 - Solutions for assignment 3
Handed out on Friday, September 27; due on Friday, October 4.
Topics: Cauchy sequences, Contractive sequences, Euclidean norm on Rn .
Problem 1.
PMath/AMath 331, Fall 2013 - Assignment 8
Handed out on Friday November 1; due on Friday November 8.
Topics: Contractions, properties of continuous functions on Rn , the Extreme value
theorem.
Problem
PMath/AMath 331, Fall 2013 - Assignment 7
Handed out on Friday October 25. Due Friday November 1.
Topics: Lipschitz functions, linear transformations, contractions.
Only submit problems 1, 2, 4, 7 and
PMath/AMath 331, Fall 2013 - Solutions for assignment 7
Handed out on Friday October 25. Due Friday November 1.
Topics: Lipschitz functions, linear transformations, contractions.
Only submit problems
PMath/AMath 331, Fall 2012 - Assignment 8
Handed out on Friday November 2; due on Friday November 9.
Topics: Contractions, properties of continuous functions on Rn , the Extreme value
theorem.
Problem
PMath/AMath 331, Fall 2013 - Solutions for assignment 2
Handed out on Friday, September 20; due on Friday, September 27.
Topics: Upper and lower bounds, Least upper bound principle, Monotone convergen
PMath/AMath 331, Fall 2013 - Assignment 3
Handed out on Friday, September 27; due on Friday, October 4.
Topics: Cauchy sequences, Contractive sequences, Euclidean norm on Rn .
Problem 1. If x1 > 0 and
Monday, September 11 Lecture 2 : Limits
Concepts: Definition of the limit L of a sequence cfw_xn in , using the definition of the limit to
determine whether a sequence converges or not in , using the
PMath/AMath 331, Fall 2017 - Assignment 1.
Posted on Friday, September 15. Due on Friday, September 22 at noon.
Topics: Definition of limit, Properties of limits, Bounded sequences, Squeeze theorem,
l
Friday, September 15 Lecture 4 : Supremum and infimum of a subset of .
Monotone sequences.
Concepts: Define the least upper bound (supremum) of a set in , define the greatest lower
bound (infimum) of
PMath/AMath 331, Fall 2017 - Assignment 7. Solutions
Posted on Friday October 27.
Topics: Lipschitz functions.
Practice problems.
Problem P1. Find a bounded continuous function on R that is not Lipsch
PMath/AMath 331, Fall 2017 - Assignment 9.
Posted on on Friday November 10; due on Friday November 17.
Topics: Intermediate Value theorem, continuity in normed vector spaces, uniformly continuous func
PMath/AMath 331, Fall 2017 - Assignment 8. Solutions
Posted on Friday November 3; due on Friday November 10.
Topics: Lipschitz constant of matrix transformations, Contractions, properties of continuou
PMath/AMath 331, Fall 2017 - Assignment 7.
Posted on Friday October 27.
Topics: Lipschitz functions.
Practice problems.
Problem P1. Find a bounded continuous function on R that is not Lipschitz. HINT:
PMath/AMath 331, Fall 2017 - Assignment 10. Solutions
Posted on Friday November 17; due on Friday November 24.
Topics: Finite dimensional vector spaces, pointwise convergence of functions, uniform con
Lecture 4: Limit of a Recursive Sequence
AMATH/PMATH 331 Winter Term 2018
Wednesday, January 10th
Text book references for additional reading:
2.6
4.1 Application of Monotone Convergence Theorem: Esti
Lecture 3: Properties of Limits
AMATH/PMATH 331 Winter Term 2018
Monday, January 8th
Text book references for additional reading:
2.52.6
3.1 Some basic properties of limits of sequences
Proposition 3.
Lecture 2: Limits
AMATH/PMATH 331 Winter Term 2018
Friday, January 5th
Text book references for additional reading:
2.32.4
2.1 The least upper bound principle
Every nonempty subset S of R that is boun
Assignment 2
AMATH/PMATH 331 Winter Term 2018
Due: Friday, January 19th at 10:30 in class
Practice problems: Not for submission.
1. Let (an )
n=1 be a sequence of real numbers. Show that if (an ) conv
Assignment 1
AMATH/PMATH 331 Winter Term 2018
Due: Friday, January 12th at 10:30 in class
Practice problems: Not for submission.
1. We defined a real number as an infinite decimal expansion of the for