Final Examination : Introduction to Real Analysis
MTH 5101
12th December, 2003
Max Credit : 200 points
Duration : 2 Hours
Notes :
1. This is an OPEN book examination. You are allowed to use only the prescribed
text. To ensure fairness, the text books are
Introduction to Analysis: Summer 2015
Practice problems - Real Number System
MTH 4101/5101
1. If r is a rational number, (r 6= 0) and x is an irrational number, prove that
r + x and rx are irrational.
Solution: Since the set of all rational numbers, Q is
Introduction to Analysis
Test - II
MTH 4101/5101
11/13/2014
Max. Credit: 50 points
Precise and complete answers are a must for full credit. Calculators are NOT
allowed. The numbers on the right indicate the maximum credit for the
corresponding question. T
Introduction to Analysis
Final Examination
MTH 4101/5101
12/11/2014
Max. Credit: 100 points
Answer all the questions. No credit will be given if only the answer is written
without showing the relevant supporting work.
(1) Refer to the proof of Theorem 4.2
Introduction to Analysis
Final Examination
MTH 4101/5101
07/08/2015
Max. Credit: 60 points
Answer all the questions. No credit will be given if only the answer is written
without showing the relevant supporting work.
(1) Refer to the proof of Theorem 4.22
Introduction to Analysis
Test - I
MTH 4101/5101
9/25/2014
Max. Credit: 50 points
Precise and complete answers are a must for full credit. Calculators are NOT
allowed. The numbers on the right indicate the maximum credit for the
corresponding question. Thr
Introduction to Analysis
Test - I
MTH 4101/5101
06/17/2015
Max. Credit: 60 points
Precise and complete answers are a must for full credit. Calculators are NOT
allowed. The numbers on the right indicate the maximum credit for the
corresponding question. Th
APPENDIX A .
LOGIC AND PROOFS
WM
Natural science is concerned with collecting facts and organizing these facts into a coherent
body of knowledge so that one can understand nature. Originally much of science was
concerned with observation, the collection
Introduction to Analysis
Final Examination
MTH 4101/5101
07/08/2015
Max. Credit: 60 points
Answer all the questions. No credit will be given if only the answer is written
without showing the relevant supporting work.
(1) Refer to the proof of Theorem 2.43
Introduction to Analysis: Fall 2015
Practice problems II
MTH 4101/5101
1. Prove that the empty set is a subset of every set.
Solution: Let A be any nonempty set. if the empty set were not to be a
subset of A there we should be able to produce an element o
Introduction to Analysis: Fall 2014
Practice problems IV
MTH 4101/5101
(1) Suppose f is a real function defined on R which satisfies
limh0 [f (x + h) f (x h)] = 0 for every x R. Does this imply that f is
continuous?
0
for x 6= 1
Solution: No. Consider the
Introduction to Analysis: Fall 2004
Practice problems VIII
MTH 4101/5101
12/12/2004
1. Discuss dierentiability at x = 0 of the function
f ( x) =
1
xm sin x if x is irrational
0
if x is rational
with m = 0, 1, 2.
Solution: When m = 0, f is not continuous a
Introduction to Analysis: Fall 2004
Practice problems VII
MTH 4101/5101
12/7/2004
1. Give an example of a non continuous function f that is dened on [a, b] and
(i) satises the intermediate value property. (ii) does not satisy the
intermediate value proper
Introduction to Analysis: Fall 2004
Practice problems III
MTH 4101/5101
10/9/2004
1. Consider the sequence cfw_an , where an =
converges to 1 .
2
1+2+3+.+n
.
n2
show that cfw_an
2. Prove that if the seqquence cfw_an converges to A, then the sequence cfw
Introduction to Analysis: Fall 2004
Practice problems II
MTH 4101/5101
9/21/2004
Read the section 1.4, 1.5, 1.6. Make sure you know how to answer the following
questions.
1. Negate the statement: For every > 0 there exists a > 0 such that whenever
x and t
Introduction to Analysis: Fall 2004
Practice problems I
MTH 4101/5101
9/13/2004
Read the section 1.1, 1.2. Make sure you know how to answer the following
questions.
1. Let Aj , j = 1, 2, . be an innite family of sets and B a set. Prove the
following.
(a)B
Introduction to Analysis: Fall 2008
Practice problems III
MTH 4101/5101
9/17/2008
1. Prove that the empty set is a subset of every set.
Solution: Let A be any nonempty set. if the empty set were not to be a
subset of A there we should be able to produce a
Introduction to Analysis: Fall 2008
Practice problems II
MTH 4101/5101
1. (a)
n
i=1
k3 =
9/10/2008
n(n+1) 2
2
.
(b) If x (0, 1) is a xed real number, then 0 < xn < 1 for all n I
N.
(c) 2n1 < n! < nn for all n = 3, 4.
(d) (Bernoullis Inequality) If x > 1,
Introduction to Analysis: Fall 2008
Practice problems - Real Number System
MTH 4101/5101
9/3/2008
1. If r is a rational number, (r = 0) and x is an irrational number, prove that
r + x and rx are irrational.
Solution: Since the set of all rational numbers,
Solutions to Exercises in
Walter Rudins
Principles of Mathematical Analysis
Third Edition
The following is work completed for the requirements of
MATH 4900 Independent Study / MATH 5210 Real Analysis II
Fall 2006 - Auburn University - Professor Greg Harri