Supplements to the Exercises in Chapters 1-7 of Walter Rudins
Principles of Mathematical Analysis, Third Edition
by George M. Bergman
This packet contains both additional exercises relating to the material in Chapters 1-7 of Rudin, and
information on Rudi
MAT337H1, Introduction to Real Analysis: Test 2 coverage
Term Test 2 will be based on the material covered in Jan 6 - Mar 1 classes, with emphasis
on the material covered in Feb 3 - Mar 1 classes (see Sections 5.1, 5.3, 5.4, 5.6, 5.7, 6.1, 6.2
of the text
MAT337H1, Introduction to Real Analysis: recommended problems for Jan 6
class
1. Explain why any rational number can be written as
least one of the numbers p, q is odd.
2. Show that if x Q, and x2 Z, then x Z.
p
where p, q Z, q 6= 0, and at
q
MAT337H1, Introduction to Real Analysis: additional recommended problems
for Feb 3 class
1. Show, using the definition of continuity, that the following functions are continuous at
every point where they are defined.
(a) f (x) = x3 .
(b) f (x) = x1 .
(c)
MAT337H1, Introduction to Real Analysis: solution of Problems 1b and 1c
from additional recommended problems for Feb 3 class
Problem 1b. Show, using the definition of continuity, that the function f (x) =
continuous at every point where it is defined.
1
x
MAT337H1, Introduction to Real Analysis: recommended problems for Jan 18
class
1. For a positive real number x = x0 .x1 . . . , let [x]n = x0 .x1 . . . xn . For two positive real
numbers x and y, we define their sum by
x + y = sup cfw_[x]n + [y]n | n Z, n
MAT337H1, Introduction to Real Analysis: recommended problems for Mar 22
class
1. Prove the following inequalities, which we used in the proof of the triangle inequality
for the uniform metric:
(a) If f, g are bounded functions on a set X, and f (x) g(x)
MAT337H1, Introduction to Real Analysis: solutions to additional
recommended problems for Mar 1 class
1. Prove the following fact which we used in the proof of Fermats theorem. Let f be a
function defined in all points of an interval (a, b) except, possib
MAT337H1, Introduction to Real Analysis: additional recommended problems
for Feb 1 class
1. Let
sn =
n
X
sin(k)
k=1
k2
.
Show that the sequence sn is convergent by showing that it is Cauchy.
2. Let an and bn be Cauchy sequences. Show that the sequence
a1
MAT 337 PROBLEM SET
Due Wednesday, March 1 at 2:10pm
RULES:
There will be a 10% penalty for problem sets submitted on March 1 after 2:10 and before
5 pm. No problem sets will be accepted after March 1, 5pm.
Students are expected to write up solutions in
MAT337H1, Introduction to Real Analysis: recommended problems for Jan 13
class
1. Show that inf cfw_x R | x > 1 = 1.
2. Let S be a bounded below set containing at least one negative number. Recall the procedure for constructing inf S for such a set. Let m
MAT337H1, Introduction to Real Analysis: recommended problems for Mar 29
class
1. In class we used that
Z b
Z b
f
(x)dx
|f (x)|dx
a
a
for a function f C[a, b]. Prove that this inequality in fact holds for any function f
Riemann integrable on [a, b]. (In
MAT337H1, Introduction to Real Analysis: solution to recommended problem
6 for Mar 15 class
Problem. Show that for a bounded function f on [a, b] the following conditions are
equivalent:
(a) f is integrable on [a, b];
(b) for any > 0 there exists > 0 such
MAT337H1, Introduction to Real Analysis: additional recommended problems
for Feb 10 class
1. Fill in the gaps in the proof of the continuity criterion in terms of sequences (first
theorem proved in Feb 10 class). Namely, explain why in the proof of the im
MAT337H1, Introduction to Real Analysis: additional recommended problems
for Feb 15 class
1. In class we proved that if f is a continuous function on [a, b], and is a number such
that f (a) < < f (b), then there is c [a, b] such that f (c) = . We defined
MAT337H1, Introduction to Real Analysis: additional recommended problems
for Mar 1 class
1. Prove the following fact which we used in the proof of Fermats theorem. Let f be a
function defined in all points of an interval (a, b) except, possibly, one point
MAT337H1, Introduction to Real Analysis: Solution to Problem 7 for Mar 22
class
Problem. Consider a sequence of continuous functions on [1, 1] given by
0, if x n1 ,
1 + nx if 1 < x 0,
n
fn (x) =
1
nx
if
0
<
x n1 ,
0, if x > n1 .
Show that fn 0 in L1 -metr
MAT337H1, Introduction to Real Analysis: solution of Problem 2 for Mar 17
class
Problem. For x > 0, define a function ln(x) by the formula
Z x
1
dt.
ln(x) =
1 t
Show, using this definition, that ln(xy) = ln(x) + ln(y). Hence show that ln is a bijection
fr
UNIVERSITY OF TORONTO
The Faculty of Arts and Science
APRIL 2017 EXAMINATIONS
MAT 337H1
Duration: 3 hours
NO AIDS ALLOWED
Total marks for this paper: 100.
This paper contains 2 pages.
Note: It is not necessary to reprove facts which are proved in the text
MAT337H1, Introduction to Real Analysis: recommended problems for Mar 15
class
1. Show that if f is integrable on [a, b], then its integral is uniquely defined. In other
words, there can exist at most one number I with the following property: For any
> 0
MAT337H1, Introduction to Real Analysis: recommended problems for Mar 31
class
1. Let f (x) be a function defined and continuous in an interval (a, b). Let also x0 (a, b),
and let t0 be any real number. Show that the initial value problem
dx
= f (x),
dt
x
MAT337H1, Introduction to Real Analysis: Quiz 2 coverage
Quiz 1 will be based on the material covered in Mar 3 - 17 classes. (See Sections 5.5, 6.3,
and 6.4 of the textbook, as well as the lecture notes on integration posted at the web page).
You will be
MAT337H1, Introduction to Real Analysis: solution to Problem 1 for Mar 24
class
Problem. Consider a sequence of continuous functions on [1, 1] given by
1
0, if x n ,
fn (x) = 12 + nx
, if n1 < x n1 ,
2
1, if x > n1 .
Show that this sequence is Cauchy in t
MAT337H1, Introduction to Real Analysis: recommended problems for Mar 24
class
1. Consider a sequence of continuous functions on [1, 1] given by
1
0, if x n ,
fn (x) = 12 + nx
, if n1 < x n1 ,
2
1, if x > n1 .
Show that this sequence is Cauchy in the L1 -
MAT 337
Sample Midterm Exam 2
NAME
NO AIDS ALLOWED
Total: 250 points, not including a bonus problem
Problem 1 [30 points]
(a) Give an example of a function f : lR lR that is continuous at every
irrational point, and that is discontinuous at every rational
Item ID: 11332
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University of Toronto
Faculty of Arts and Sciences
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Sample Final Exam, April-May 2014
MAT 337 H1
Intro Real Analysis
.c
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Instructor: Regina Rotman
Duration - 3 hours
No aids allowed
Total marks for this paper is 400
Please wr
MAT337, Midterm 1
Sample
w Answer
ww
.
Note: This is a sample of answers(or ideas) to your first midterm exam.
there are several correct answers, but I will only write one.
Problem 1. [20 points]
Determine which of the following sequences converge:
1
1
(a
Exercises for Section 1.3
1. Prove the converse of the Generalized Heine-Borel: A compact subset
of Rn is closed and bounded.
Proof. Let A be a compact set in Rn . To see that A is bounded,
consider the open cover
O = cfw_B (0, r) : r R ,
n
O
of all open
MAT337H1, Introduction to Real Analysis: notes on Riemann
integration
1
Definition of the Riemann integral
Definition 1.1. Let [a, b] R be a closed interval. A partition P of [a, b] is a finite set of
points x0 < x1 < < xn1 < xn in [a, b] such that x0 = a
MAT337H1, Introduction to Real Analysis: solution to Exercise C for
Section 5.6 and Problem 1 from additional recommended problems for Feb 15
class
Exercise C for Section 5.6. Show that 2 sin(x) + 3 cos(x) = x has three solutions.
Solution. Consider the f
MAT337, Real Analysis
Midterm 1
Solutions
1. (a) Give a definition of a Cauchy sequence.
A sequence an os real numbers is called Cauchy if for any real number
> 0 there exists a natural number N such that |an am | < as long as
m, n N .
(b) Let an be a Ca
MAT337, Real Analysis
Midterm 2 Solutions
1. (a) Let f be a function on (a, b), and let x0 (a, b). Let also L R. Define
what it means that limxx0 f (x) = L.
Solution. We say that limxx0 f (x) = L if for any > 0 there exists
> 0 such that for any x (a, b)
Pure Math 450/650 Winter 2017
Fourier Series and Lebesgue Measure
Professor: Kenneth Davidson
Email: [email protected]
Office: MC 5324
Class meets: MWF 9:3010:20 in MC 4045
Website: www.math.uwaterloo.ca/krdavids/PM450.html
Textbooks: These will be av