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: 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,