5615H HW3
1. Prove the following statements:
(i) every real number is a limit point of Q
(ii) a nite union of closed sets is closed
(iii) if S is a (nonempty) closed and bounded subset of R, then sup
5615H HW11
1. A function f : (a, b) R is called uniformly dierentiable if it is dierentiable and for each
> 0, there is > 0 such that x, y (a, b) and 0 < |x y| < imply
f (y) f (x)
f (x) < .
yx
Prove
5615H HW1
1. Prove that
3 Q.
/
Solution. Suppose that 3 = m/n with m, n N having no prime factors in common. Then
3 = m2 /n2 so 3n2 = m2 , which shows m2 is divisible by 3. By Euclids lemma m is also
5615H HW8
1. Prove that the Cauchy product of two absolutely convergent series is absolutely convergent.
Solution. Let
B. Dene
n=0 an
n=0 bn
and
be absolutely convergent, say
n=0 |an |
= A and
n=0 |bn
SECOND PROBLEM SET
Math 5615H: Honors Analysis
Due W 20 September, 2017.
10 points each; total 50 points.
1. Let F be a field. Show that there exist at most two solutions of x2 = 1.
True/false: Is it
SIXTH PROBLEM SET
Math 5615H: Honors Analysis
Due W 18 October, 2017.
12 points each; total 60 points.
1. If a1 = 1, and
an+1 =
1
for n = 1, 2, . . . ,
1 + an
prove that the sequence cfw_an converges
MATH 5615H: INTRODUCTION TO ANALYSIS I
SAMPLE MIDTERM EXAM I
You may not use notes, books, etc. Only the exam paper, a pencil
or pen may be kept on your desk during the test. Calculators are not
allow
THIRD PROBLEM SET
Math 5615H: Honors Analysis
Due W 27 September, 2017.
10 points each; total 50 points.
1. Let A := cfw_a1 , a2 , . . . be a set of real numbers defined as follows:
a1 = 1,
and ak+1
Math 5615H: Introduction to Analysis I.
Fall 2017
Homework #4, due Weds. October 4. 50 points.
#1. Let f be a mapping of A to B. Show that for each B1 B and B2 B,
their inverse images satisfy the prop
FIRST PROBLEM SET
Math 5615H: Honors Analysis
Due W 13 September, 2017.
10 points each; total 60 points.
1. Prove that
10 and
5
2 are not rational numbers.
2. Let A and B be bounded sets in R. Conside
5615H HW10
1. Let f : [a, b] R. Prove the following statements:
(i) If f is continuous and injective, then f is monotone.
(ii) If f is dierentiable and f (x) = 0 for all x (a, b), then f is injective.
5615H HW4
1. Let X be a metric space and suppose Y X. If E Y , we say E is open relative to Y if
for each x E, there is a neighborhood B (x) of x such that B (x) Y E.
Let X = R R, Y = R cfw_0, and E =
Math 5615H Midterm 3
Name:
Instructions:
This exam contains 5 pages, with one question per page. Each question is worth 20 points. The
exam is worth 100 points in total.
1
1. Let f : [0, 1] [0, 1] be
Math 5615H Midterm 2
Name:
Instructions:
This exam contains 4 questions. Each question is worth 25 points. The exam is worth 100
points in total.
1
1. Let cfw_an be a sequence in a metric space X. Pr
5615H HW2
1. Let f : S T be a function.
(a) Formulate a condition which is equivalent to injectivity of f , such that your condition
incorporates the image and preimage of f as well as all subsets of
Math 5615H Midterm 1
Name:
Instructions:
This exam contains 5 questions. Each question is worth 20 points. The exam is worth 100
points in total.
1
1. Let S be a set, and let T be the set of all funct
5615H HW7
1. Let cfw_an be a sequence with an 0 for all n. If
an converges, must it be true that
2n a2n also converges? Prove it, or provide a counterexample.
Solution. No. Let an = 1/n if n = 2k for
5615H HW5
1. The Cantor set consists of real numbers which admit a ternary (base 3) decimal expansion
of the form .a1 a2 a3 . where an cfw_0, 2 for all n. Use this to prove that the Cantor set is
unco
5615H HW6
1. Let cfw_an and cfw_bn be convergent sequences in Rd . Prove that then then lim(an + bn ) =
lim an + lim bn . Prove also that if c R then lim(c an ) = c lim an .
Solution. Assume lim an
5615H HW9
1. Let f : [a, b] R be a continuous function such that f (a) < 0 < f (b). Prove that there
exists c (a, b) such that f (c) = 0.
Solution. Let S = cfw_x [a, b] : f (x) < 0 and c = sup S. If c
5615H HW12
1. Dene f : [0, 1] R by f (0) = 0 and
f (x) =
0,
xQ
/
1/n, x = m/n, where m, n N have no common prime factors
Prove that f R and
1
f dx = 0.
0
Solution. Fix > 0. Note that S = cfw_x [0, 1]
FIFTH PROBLEM SET
Math 5615H: Honors Analysis
Due W 11 October, 2017.
10 points each; total 50 points.
1. Show that for an arbitrary set E in a metric space (X, d), the set E 0 of its
limit points is