FOUNDATIONS OF ANALYSIS
SPRING 2012
EXAM #1
Directions.
Please work as many problems as you can on the enclosed pages. The reader
will read all of the problems and assign a numerical. grade between 0 and
20. to each problem. The scores of the ﬁve problems

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 02
Solution Published February 5, 2016
5 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborat

FOUNDATIONS OF ANALYSIS
Homework 24
Chris Pontiff
[email protected]
April 19, 2016
Problem.
A. Prove that every convergent sequence in a metric space is a Cauchy
sequence.
B. Provide an example of a Cauchy sequence, in a metric space, that is
not convergent.

FOUNDATIONS OF ANALYSIS
Homework 20
Chris Pontiff
[email protected]
April 1, 2016
Problem. Let cfw_ak
k=1 denote a sequence in cfw_0, 1, 2. Prove that
converges to a real number in [0, 1].
P
Proof. It suffices to show(i)
k=1
[0, 1].
ak
3k
P
ak
k=1 3k
conve

FOUNDATIONS OF ANALYSIS
Homework 22
Chris Pontiff
[email protected]
April 12, 2016
Problem. Let u and v denote vectors in an inner-product space, V (over
the real field, R). Prove that hu, vi = 0 if and only if
|u| |u + v|
for all scalars, .
Proof. It suffic

FOUNDATIONS OF ANALYSIS
Homework 19
Chris Pontiff
[email protected]
March 31, 2016
Problem. Prove that the additive inverse of an integer is an integer. (Note
that this is an induction proof.
Proof. Argue that if k Z and (k) Z then l Z whenever l Z and
l k.

FOUNDATIONS OF ANALYSIS
Homework 26
Chris Pontiff
[email protected]
April 25, 2016
Recall that a point x in a metric space, M , is a limit point of a subset S
of M , if every open ball containing x contains an infinite number of points
of S. (This is equival

FOUNDATIONS OF ANALYSIS
Homework 21
Chris Pontiff
[email protected]
April 5, 2016
Problem. Let V denote a vector space over a field F .
A. Prove that 0v = o, v V without using the fact that (1)w =
w, w V .
B. Prove that (1)v = v, v V .
Proof. Proof for part

FOUNDATIONS OF ANALYSIS
Homework 20
Chris Pontiff
[email protected]
April 1, 2016
Problem. Let cfw_ak
k=1 denote a sequence in cfw_0, 1, 2. Prove that
converges to a real number in [0, 1].
P
Proof. It suffices to show(i)
k=1
[0, 1].
ak
3k
P
ak
k=1 3k
conve

FOUNDATIONS OF ANALYSIS
Homework 27
Chris Pontiff
[email protected]
April 28, 2016
Recall the following definition. Let S denote a subset of a metric space
(M, d). A point x0 M is a boundary point of S if for every r > 0 the
ball, B(x0 , r), intersects S and

FOUNDATIONS OF ANALYSIS
Homework 25
Chris Pontiff
[email protected]
April 25, 2016
Recall the following definition. Let x0 denote a point in a metric space,
(M, d) and let r > 0. The closed ball of radius r centered at x0 , denoted
here by B(x0 , r) is defin

FOUNDATIONS OF ANALYSIS
Homework 14
Chris Pontiff
[email protected]
March 22, 2016
Problem. Let D denote a subset of R and let x0 D. Let f and g be realvalued functions defined on D, both continuous at x0 . Recall that the product of f and g, denoted here by

FOUNDATIONS OF ANALYSIS
Homework 18
Chris Pontiff
[email protected]
March 28, 2016
Recall that the Fibonacci sequence cfw_F (n)
n=0 is defined by
F (0) = 0
F (1) = 1
F (n) = F (n 1) + F (n 2), n = 2, .
Problem. Prove that
n
X
F (2i) = F (2n + 1) 1, n = 0, 1,

FOUNDATIONS OF ANALYSIS
Homework 16
Chris Pontiff
[email protected]
March 25, 2016
Recall that if A is a non-empty subset of Zn then A has cardinality m for
some m Z+ , m n.
Problem. Prove the following proposition. (using the definition of a finite
set from

FOUNDATIONS OF ANALYSIS
Homework 17
Chris Pontiff
[email protected]
March 28, 2016
Recall that if m Z then every integer greater than m can be written in
the form m + n for some n Z+ .
Problem. Let F denote a field and let ak F, k = 1, . Let m, p denote
a po

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 03
Solution Published February 5, 2016
5 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborat

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 06
Solution Published on February 19, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collab

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 05
Solution Published February 9, 2016
5 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborat

Homework 2
There is only one homework problem. The expectation is that the proof will be very high quality.
Let f : A B and suppose C A and D B.
Prove that f (C) D if and only if C f 1 (D).

Homework 4
There are only two homework problems. The expectation is that the proofs will be very high quality.
1. Use Mathematical Induction to prove that
|x1 + x2 + x3 + + xn | |x1 | + |x2 | + + |xn |
for all n N.
2. Suppose that for all > 0 we have 0 x

Homework 1
There is only one homework problem. The expectation is that the proof will be very high quality.
Let A and B be subsets of a universal set U. Prove U \ (A \ B) = (U \ A) B.

Homework 5
There are only two homework problems. The expectation is that the proofs will be very high quality.
1. Let f : A R and g : R R be functions. Suppose that there exists an M R such that |f (x)| M
and |g(x)| M for all x A (so f and g are bounded f

Homework 3
There is only one homework problem. The expectation is that the proof will be very high quality.
Exercise 9, section 2.4.
Hints:
Repeatedly use (without proof) the result of part d of Example 2.4.11
A polynomial of degree n has at most n rea

FOUNDATIONS OF ANALYSIS
Math 4090
SPRING 2016
SYLLABUS
INSTRUCTOR
Maya Kiehl
Amos Eaton 326
518-276-2307
[email protected]
Workshops: Monday 8:30 - 10 AM
Wednesday* 10 - 11:30 AM
in AE 326
or by appointment
*cancelled some weeks
TEACHING ASSISTANT
Jared Farr

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 12
Solution Published on March 4, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborat

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 11
Solution Published on March 4, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborat

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 13
Solution Published on or after March 8, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a c

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 10
Solution Published March 1, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborative

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 09
Solution Published March 1, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collaborative

FOUNDATIONS OF ANALYSIS
SPRING 2016
HOMEWORK 15
Solution Published on March 22, 2016
7 points
Directions. Please submit your answer to the following problem in a LATEXprepared document. Class participants are encouraged to prepare solutions
in a collabora