September 2223, 2015
MATH2201/MATH2241 Intro. to Math. Analysis
Tutorial 2
Problem 1. Suppose that S R and = sup(S) belongs to S. If t S, then show that
/
sup(S cfw_t) is the larger of and t.
Solution. Set T := S cfw_t. If t , then t is an upper bound for
September 2223, 2015
MATH2201/MATH2241 Intro. to Math. Analysis
Tutorial 2
Problem 1. Suppose that S R and = sup(S) belongs to S. If t S, then show that
/
sup(S cfw_t) is the larger of and t.
Problem 2. Let S R be non-empty. Prove that R is the supremum o
September 16, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Homework 2
Problem 1. (25 points) Prove that if 0 < x < for some , x R, then x3 < 3 .
Solution. By equation (2.3) in the lecture notes, if 0 < x < , then x2 < 2 . We next note
that by (F5),
x2
September 23, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Homework 3
Special note: The assignment is due on Friday at 11:00 due to the holiday on Thursday.
Problem 1. (25 points) The 3-adic rationals are dened by 3a : a Z, n N0 . Prove that
n
the 3-ad
September 9, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Homework 1
Problem 1. (25 points) Show that for a eld F, we have (1)1 = 1.
Solution. Recall that by Proposition 2.4 (2), we know that (1)1 is the unique b such that
(1) b = 1. Hence it suces to
September 30, 2015
MATH2201/MATH2241 Intro. to Math. Analysis
Homework 4
Problem 1. (25 points) Using the denition, show that for any a, b, c R, b = 0, we have
an2 + n
a
= .
2+c
n bn
b
lim
Problem 2. (25 points) Suppose that (an )nN is a sequence of real
September 9, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Homework 1
Problem 1. (25 points) Show that for a eld F, we have (1)1 = 1.
Problem 2. (25 points) Suppose that F is a set together with two binary operators + and
which satisfy all of the eld a
September 16, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Homework 2
Problem 1. (25 points) Prove that if 0 < x < for some , x R, then x3 < 3 .
Problem 2. (25 points) Prove that an ordered eld cannot be a nite set.
Problem 3. (25 points) Prove that fo
Midterm Exam 1
Math 2201/2241 (2014-2015, Semester 1)
Dr. Benjamin Kane
Monday, October 6, 2014
You have one hour and 50 minutes for the exam. There are 4 questions (25 points each). There
are a total of 12 pages (6 pages front and back including this tit
September 1516, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Tutorial 1
Example. Here are a few examples of elds. Check that they satisfy all of the axioms.
(1) The set Z/5Z = cfw_0, 1, 2, 3, 4 of integers modulo 5 with the usual addition and multiplic
September 1516, 2014
MATH2201/MATH2241 Intro. to Math. Analysis
Tutorial 1
Example. Here are a few examples of elds. Check that they satisfy all of the axioms.
(1) The set Z/5Z = cfw_0, 1, 2, 3, 4 of integers modulo 5 with the usual addition and multiplic
September 2930, 2015
MATH2201/MATH2241 Intro. to Math. Analysis
Tutorial 3
Problem 1. Construct a sequence of nested unbounded intervals whose intersection is trivial.
Problem 2. For R, set
S := cfw_x R : x < + > 0
and
S := cfw_x R : x + > 0 .
Prove that
MATH3603/4
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Probability Theory
Exercise 4
Let X1 , X2 . . . . be independent and identically distributed continuous random variables
with common probability distribution function F and density function f =
MATH3603/3
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Probability Theory
Exercise 3
1. Suppose that X is a random variable with expectation 10 and variance 15. What
can we say about P(5 < X < 15)?
Solution.
P(5 < X < 15) = P(5 < X 10 < 5) = 1 P(|X
MATH2201/MATH2241
Introduction to Mathematical Analysis
Dr. Ben Kane
Department of Mathematics
The University of Hong Kong
First Semester 2015-2016
Date: September 1, 2015
1
2
1. Practical Information
Instructor
Dr. Ben Kane
Email: bkane[at]maths.hku.h
Midterm Exam 1 Makeup
Math 2201/2241 (2014-2015, Semester 1)
Dr. Benjamin Kane
Monday, October 24, 2014
You have one hour and 50 minutes for the exam. There are 4 questions (25 points each). There
are a total of 12 pages (6 pages front and back including
Midterm Exam 1
Math 2201/2241 (2014-2015, Semester 1)
Dr. Benjamin Kane
Recall the axioms for an ordered eld:
(F1) For all a, b S, we have a + b = b + a (commutative law for addition).
(F2) For all a, b S, we have a b = b a (commutative law for multiplica
Midterm Exam 1
Math 2201/2241 (2014-2015, Semester 1)
Dr. Benjamin Kane
Recall the axioms for an ordered eld:
(F1) For all a, b S, we have a + b = b + a (commutative law for addition).
(F2) For all a, b S, we have a b = b a (commutative law for multiplica
MATH2201 Introduction to Mathematical Analysis
Assignment 6
Due date: 15 April, 2011
1. Let f be dened on R, and suppose that |f (x) f (y )| (x y )2 for all x, y R. Prove
that f is a constant function.
1
2. Let f (x) = x2 sin( x ) +
x
2
for x = 0 and f (0
MATH2201 Introduction to Mathematical Analysis
Assignment 5
Due date: 1 April, 2011
1. Let S R and suppose there exists a sequence (xn ) in S that converges to a number
x0 S .
/
(a) Show that there exists an unbounded continuous function on S .
(b) Show t
MATH2201 Introduction to Mathematical Analysis
Assignment 4
Due date: 18 March, 2011
1. Prove that each of the following functions is continuous by verifying the - property.
(a) f (x) = sin x
(b) g (x) = x3
(c) h(x) = x
2. (a) Let f (x) = 1 for rational n
MATH2201 Introduction to Mathematical Analysis
Assignment 3
Due date: 25 February, 2011
1. (a) Show that
lim sup(sn + tn ) lim sup sn + lim sup tn
for bounded sequences (sn ) and (tn ).
Hint : First show supcfw_sn + tn : n > N supcfw_sn : n > N + supcfw
MATH2201 Introduction to Mathematical Analysis
Assignment 2
Due date: 11 February, 2011
1. Determine the limits of the following sequences, and then prove your claims using the -N
property.
(a) an =
n
n2 +1
(b) bn =
2n+4
5n+2
(c) cn =
sin n
n
2. Let (tn )