MA203
Real Analysis
Lecture Notes
Eleni Katirtzoglou
c Eleni Katirtzoglou London 2015
Table of Contents
Table of Contents
iv
1 Sequences and Series in R
1
1.1
Sequences of Real Numbers . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Series . . . . . .

MA203 Assignment 1
2015 2016
Question 1.1: (a) State clearly what does it mean to say that a sequence (xn )
in R is Cauchy.
(b) Give a pictorial illustration of a Cauchy sequence.
(c) Is the sequence (1)n ) Cauchy? Explain by either using the denition or

MA203 Assignment 2
2015 2016
Question 2.1: Let (xn ) be a sequence in R. State clearly what does it mean to
say that the series xn converges.
n=1
Question 2.2: Solve Exercise 1.7 from the lecture notes.
Question 2.3: Solve Exercise 1.18 from the lecture n

MA203 Assignment 3
2015 2016
Question 3.1: (a) Let (X, d) be a metric space and E X. State carefully
what does it mean to say that the set E is open.
(b) Give a graphical illustration of V1 (3, 0) in (R2 , | | ).
+
(c) Consider the points x = (2, 3) and y

MA203 Assignment 4
2015 2016
Question 4.1: (a) Let (X, d) be a metric space and E X. What does it mean
to say that x X is a boundary point of E?
(b) Is it possible to have bd(E) E = ? If yes give an example. If not give a
counterexample.
Question 4.2: Let

MA203 Assignment 5
2015 2016
Question 5.1: Consider the following statement:
Let (X, d) be a metric space and E X. Then cfw_X is an open nite
cover for E and so E is compact.
Is the statement true or false? Explain.
Question 5.2: Use Denition 2.47 to show

MA203 Assignment 6
2015 2016
Question 6.1: Let (X, dX ) and (Y, dY ) be metric spaces and let c X.
(a) What does it mean to say that a function f : X Y is discontinuous at c?
(b) Give a graphical illustration of your answer in part (a).
Question 6.2: Let

MA203 Assignment 7
2015 2016
Question 7.1: Let (X, dX ) and (Y, dY ) be metric spaces and consider the
statement:
Let f : X Y be continuous. If f : X Y is uniformly continuous then X is
compact.
Is this statement true or false? If it is true give a brief

MA203 Assignment 8
2015 2016
Question 8.1: Let f : (a, b) R be differentiable and let c (a, b). Consider
the statement:
If f 0 (c) < 0 then there is > 0 such that f (c) < f (x) for all x (c , c).
Is this statement true or false? If it is true give a graph

MA203 Assignment 9
2015 2016
Question 9.1: (a) Let f : [a, b] R be a bounded function. State clearly an
integrability criterion that also results in the calculation of the integral.
(b) Let f : [a, b] R be an integrable function. What is the indenite inte

MA203 Round Table class 3 2015—2016
Quiz—Answers
Question 1: Which of the following statements is true?
A. The interval (1, 5) is an open set and so it is not a closed set in (R, | -
B. Let (X, 61) be a metric space. If E g X is not open then E is closed

MA203 Round Table class 7
2015 2016
Quiz-Answers
Question 1: Let (X, dX ) and (Y, dY ) be metric spaces. Which of the following
statements implies that the function f : X Y is not uniformly continuous?
There maybe more than one correct answers.
A. There i

MA209 Assignment 1
2015 2016
Solutions
Solution to Question 1.1: [8 marks] Distribution: (a)= 2, (b)= 3, (c)= 3.
(a) A sequence (xn ) in R is said to be a Cauchy sequence if for every > 0 there
is N N such that
|xn xm | < for all n, m > N.
(b) Diagram in

MA203 Assignment 5
2015 2016
Solutions
Solution to Question 5.1: [4 marks] False. If it was true then every metric
space would have been compact. Compactness requires to nd a nite subcover
of any open cover. In other words to be able to reduce the innite

MA203 Assignment 4
2015 2016
Solutions
Solution to Question 4.1: [5 marks] Distribution of marks: (a)= 2 and (b)=3.
(a) A point x X is said to be a boundary point of E if V (x) E = and
V (x) E c = for all > 0.
(b) Yes, consider for example the set E = (0,

MA209 Assignment 2
2015 2016
Solutions
Solution to Question 2.1: [2 marks] Let (sn ) be the sequence of partial sums
of the series xn . That is, sn = n xn for any n N. We say that the
n=1
k=1
series xn converges if and only if the sequence (sn ) converges

MA209 Assignment 3
2015 w 2016
Solutions
Solution to Question 3.1: [10 marks] Distribution of marks: (a): 2: (b)=3
and (C)=5.
(a) The set E is open if for every x E E there is E > 0 such that (_: E.
(13} Let ($1,562) 6 R3. Then ($1,362) E V1d°°(3,0) if
“(

MA203 Assignment 8
2015 — 2016
Solutions
Solution to Question 8.1: [3 marks] Time. For a formal proof follow the
method on Lemma 4.4. A graphical illustration is given below.
Solution to Question 8.2: [3+3:6 marks]
(a) Suppose on the contrary that there

MA203 Assignment 6
2015 — 2016
Solutions
Solution to Question 6.1: [2+2=4 marks]
(a) The function f is discontinuous at c E X if there is e > 0 such that for all
(5 > 0 there is m E X with dx($,c) < 5 but dy(f($)1f(6)2 e.
(b) X
Solution to Question 6.2:

MA203 Assignment 9
2015 2016
Solutions
Solution to Question 9.1: [2+3=5 marks]
(a) Let f : [a, b] R be a bounded function. If (Pn ) is a sequence of partitions
of [a, b] such that
lim (U (f, Pn ) L(f, Pn ) = 0,
n
then f is integrable and
b
f = lim (L(f, P

MA203 Round Table class 5
2015 2016
Quiz-Answers
Question 1: Let (X, d) be a metric space and E X. Which of the following
statements means that E is not compact? There maybe more than one correct
answers.
A. E is not closed.
B. There is a sequence in E wi

MA203
REAL ANALYSIS
NOTES
Adapted from the 2015 course pack and lecture slides
Prepared by Luke Milsom
APRIL 2015
Chapter 1
Distance, limits and
continuity.
1.1
1.1.1
Metric spaces
Distance
The concept of distance is a very important one and underpins muc