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 mat
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
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,
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 t
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 (
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 d
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
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 def
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
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
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
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 hold
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 inte
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 Fe
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),
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 p
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
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
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
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
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 numbe
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 not
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 ,
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
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
irrati
Item ID: 11332
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w.
University of Toronto
Faculty of Arts and Sciences
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a
Sample Final Exam, April-May 2014
MAT 337 H1
Intro Real Analysis
.c
om
Instructor: Regina Rotman
Duration - 3 hours
No a
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]
Dete
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 t