2
Feedback
2E Intro to real analysis 2013/14
.
Feedback and solutions
Q1
Let A, B be subsets of R such that inf A and sup B exists and
inf A > 0. Dene
C=
1
+ b : a A, b B .
a
Explain why this makes sense (i.e. why a = 0 for every a A), and
prove that sup
Ofce Hours: 13:3014:30 Mon, Tues, Thus in maths 308 1
Q1
Find K R so that the implication
 x + 2 1 =
2
Ex Sheet
2E Intro to real analysis 2013/14
.
Please send comments, corrections,
questions and clarications to me via
stuart.white@glasgow.ac.uk
1
x+5
Ofce Hours: 13:3014:30 Mon, Tues, Wed, in maths 308 1
The third webassign assignment will be released on Friday 31 Jan
and the assignment must be completed by 03:00 on Tuesday 4 Feb.
The assignment will be mainly based on the questions below, mainly
in a
Solutions and Comments 1 2 3
Q1
.
Please send comments, corrections,
questions and clarications to me via
stuart.white@glasgow.ac.uk
1
Find K R so that the implication
 x + 2 1 =
2
Solutions
2E Intro to real analysis 2013/14
x+5
+ 1 K  x + 2
x1
is tru
Thursday, 28th February, 2013
10:10 a.m. to 10:55 p.m.
University of Glasgow
School of Mathematics and Statistics
Mathematics 2E  Introduction to Real Analysis
Class Test
An electronic calculator may be used provided that it does not have
a facility for
Ofce Hours: 13:3014:30 Mon, Tues, Wed, in maths 308 1
The rst webassign assignment will be released on Friday 17 Jan
and the assignment must be completed by 03:00 on Tuesday 21 Jan.
The assignment will be mainly based on the questions below, mainly
in a
Thursday, 1st March, 2012
10:10 a.m. to 10:55 a.m.
University of Glasgow
School of Mathematics and Statistics
Mathematics 2E  Introduction to Real Analysis
Class Test
Candidates should attempt all questions.
1. Let f : R R be a function. Consider the sta
2E: Proofs of Theorems 3.2.3 and 3.2.9
Please send comments, clarications and report errors to me via
stuart.white@glasgow.ac.uk
Theorem 3.2.3
1
n=1 n converges if and only if n > 1.
We have already seen that the harmonic series n1 diverges. For 0 < 1, we
2E: Comments on the 201112 Class test
Please send comments, clarications and report errors to me via
stuart.white@glasgow.ac.uk
General Comments
I wrote this document after marking the 201112 class test to try and indicate you students
could improve. Th
Handout
2E Intro to real analysis 2013/14
.
3 Sequences
In this chapter we will start our study of limits in the context
of sequences. A real sequence is an innite list of real numbers,
like 1, 1/2, 1/3, 1/4, . . . . To make this precise we make the follo
Solutions and Comments 1 2 3
Q1
Please send comments, corrections,
questions and clarications to me via
stuart.white@glasgow.ac.uk
3n2 5n + 2
2n2 + 6n + 1
The point is that there are only nitely many values of n N with
n < n0 , namely n = 1, 2, . . . , n0
Solutions and Comments 1 2 3
1
Solutions
2E Intro to real analysis 2013/14
.
Please send comments, corrections,
questions and clarications to me via
stuart.white@glasgow.ac.uk
1
Q1
Let P, Q and R be statements. Construct a truth table for
(not( P or Q) an
Handout
2E Intro to real analysis 2013/14
.
Order, bounds, and axioms for the real numbers
The ordered eld axioms
What are the real numbers anyway? Weve been working with
them for some time, but without having discussed what it means to be
a real number,
Appendix C
Which Test for Convergence?
Given
an .
No
Does an 0 ?


an diverges.
Yes (or dont know)
?
Is an 0 (eventually) ?
Yes
No
?
?
Do the an s
(eventually) alternate
+ + + ?
Does an have
n!, xn , nn , or similar,
as a factor ?
Yes
No
?
Try the
Ratio
Solutions and Comments
Q1
1 2
Find K R so that the implication
x +5
 x + 2 1 =
+ 1 K  x + 2
x1
is true for all x R. As always, make sure you fully justify your answer.
This exercise is very similar to example 1.8 from lectures. The first
5
step is t
Solutions and Comments
1 2
.
If youve not seriously tried these exercises, please dont look at these solutions
and comments, until you have. Youll
get the most benefit from reading these
comments, when youve first thought
hard about them yourself, even if
1
Feedback
2E Intro to real analysis 2013/14
.
Feedback and solutions
Q1
Find a value of K > 0 such that the implication
 x 3 < 1 =
x2 + 2x + 5
4 K  x 3
x+2
is true. Make sure you justify your choice of K, with a proof of the
implication for your val
Ofce Hours: 13:3014:30 Mon, Tues, Thus in maths 308 1
The tenth and nal webassign assignment will be released on
Friday 21 Mar and the assignment must be completed by 03:00 on
Tuesday 24 March. The assignment will be mainly based on the
questions below,
Solutions and Comments 1 2 3
.
Please send comments, corrections,
questions and clarications to me via
stuart.white@glasgow.ac.uk
1
Q1
Let f , g be real functions which are continuous at c and let
R.4
a) Prove directly from the denition that f (which is
Feedback
2E Intro to real analysis 2013/14
4
.
Feedback
Q1
Let f : R \ cfw_ 3 R be given by f ( x ) =
2
from the denition, that f is continuous at 2.
x 2 + x +2
2x 3 .
Show, directly
This question was similar to a load of examples weve done in
lectures,
Wednesday, 7th May, 2014
09.30 am to 11.00 am
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2E  Introduction to Real Analysis
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display
Specimen Paper
Duration: 45 mins
University of Glasgow
Department of Mathematics
Mathematics 2E  Introduction to Real Analysis
Class Test
Candidates should attempt all questions.
1. Let f : R R be a function. Consider the statement
P :
> 0, M R such tha
Ofce Hours: 13:3014:30 Mon, Tues, Thus in maths 308 1
The eighth webassign assignment will be released on Friday 7 Mar
and the assignment must be completed by 03:00 on Tuesday 10 March.
The assignment will be mainly based on the questions below, mainly
i
Solutions
7
2E Intro to real analysis 2013/14
Solutions and Comments 1 2 3
Q1
3n 2n
3n +1 ,
n =1
2n 5
3n + 1 ,
n =1
Please send comments, corrections,
questions and clarications to me via
stuart.white@glasgow.ac.uk
1
Show that each of the following seri
?, ?th ?, 2012
? to ?
University of Glasgow
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2E  Introduction to Real Analysis
Candidates must attempt all questions.
1. (i) Consider the statement
P :
x [1, 3], y [2, 4] such that x2 3x + 4 < y.
W
2E 201213: Solutions
1. (i) D (ii) C (iii) D (iv) A (v) A
2. (i). Suppose that xn L for some L R as n and note that for n even, xn = 3
and for n odd xn = 1. Take = 2 > 0 in the denition of convergence, and so there
exists n0 N such that xn L < whenever
Thursday, 4th March, 2010
12.05 p.m. to 12.50 p.m.
University of Glasgow
Department of Mathematics
Mathematics 2E  Introduction to Real Analysis
Class Test
An electronic calculator may be used provided that it does not have
a facility for either textual