Math 320-3: Homework 5 Solutions
Northwestern University, Spring 2015
1. Wade, 12.2.7. Suppose that Q = cfw_(x, y) R2 | x > 0 and y > 0 and that f is a continuous
function on R2 whose rst-order partial derivative with respect to x satises |fx | 1. (So, we
Math 320-3: Homework 4 Solutions
Northwestern University, Spring 2015
1. Wade, 12.1.5acd. Let E be a Jordan region in Rn .
(a) Prove that E o and E are Jordan regions.
(c) Prove that Vol(E) > 0 if and only if E o = .
(d) Let f : [a, b] R be continuous on
Math 320-2: Homework 3 Solutions
Northwestern University, Winter 2015
1. Wade, 7.1.7. Suppose that f is uniformly continuous on R. If yn 0 as n and
fn (x) := f (x + yn ) or x R. Prove that fn converges uniformly on R.
Proof. For a fixed x R, since yn 0 an
Math 320-2: Homework 3
Northwestern University, Winter 2015
1. Wade, 7.1.7. Suppose that f is uniformly continuous on R. If yn 0 as n and
fn (x) := f (x + yn ) or x R. prove that fn converges uniformly on R.
2. Wade, 7.2.5. Show that
X
x
1
f (x) =
sin
k
k
Math 320-2: Homework 5
Northwestern University, Winter 2015
1. Suppose that U and V are open subsets of R. Show that U V is open in R2 . Here we are
considering R with the standard metric, but feel free to use whichever of the Euclidean, taxicab,
or box m
Math 320-2: Homework 2
Northwestern University, Winter 2015
1. Determine whether or not each of the following sequences of functions is pointwise convergent
on the specied domain.
(a) fn (x) = sin(xn + 1)
xn2 + x
on [1, 1]
n2 + 1
(b) gn (x) = (x 1)n
2 +
Math 320-2: Homework 1
Northwestern University, Winter 2015
1. Suppose that an 0 for all n and that (bn ) is a bounded sequence. If
an converges, show
that
an bn converges. Also, give an example showing this is not necessarily true without the
assumption
Math 320-2: Midterm 1 Practice
Northwestern University, Winter 2015
1. Give an example of each of the following. No justication is needed.
(a) A series of numbers which is convergent but not absolutely convergent.
(b) A sequence of functions fn which conv
Math 320-2: Homework 1 Solutions
Northwestern University, Winter 2015
1. Suppose that an 0 for all n and that (bn ) is a bounded sequence. If
an converges, show
that
an bn converges. Also, give an example showing this is not necessarily true without the
a
Math 320-3: Midterm 2 Practice Solutions
Northwestern University, Spring 2015
1. Give an example, with justication, of each of the following.
(a) A Jordan region E R2 which contains a non-Jordan region E0 E.
(b) An integrable function on [0, 1] [0, 1] whi
Math 320-3: Quiz 5 Solutions
Northwestern University, Spring 2015
Define f : [0, 1] [0, 1] R by
(
1
f (x, y) =
xy
Show that the iterated integrals
Z 1Z 1
Z
f (x, y) dx dy
0
0
if x = y
if x =
6 y.
1Z 1
f (x, y) dy dx
and
0
0
exist and are equal. You should
Math 320-3: Quiz 4 Solutions
Northwestern University, Spring 2015
Dene f : [0, 1] [0, 1] R by
f (x, y) =
x
0
x=y
x = y.
In other words, at points (x, x) on the diagonal of the unit square f has the value x and elsewhere
it has the value 0. Construct an ex
Math 320-2: Homework 2 Solutions
Northwestern University, Winter 2015
1. Determine whether or not each of the following sequences of functions is pointwise convergent
on the specified domain.
r
2+x
1
xn
on [1, 1] (b) gn (x) = (x 1)n 2 + x3 + x cos on [1,
Math 320-2: Homework 6
Northwestern University, Winter 2015
1. Give counterexamples to each of the following false statements. Provide brief justication.
(a) Any closed and bounded subset of R\Q with respect to the Euclidean metric is compact.
(b) Any con
Math 320-2: Final Exam Practice Solutions
Northwestern University, Winter 2015
1. Give an example of each of the following. No justication is needed.
(a) A closed and bounded subset of C[0, 1] which is not compact.
(b) An unbounded subset of R3 which is n
Math 320-2: Final Exam Practice
Northwestern University, Winter 2015
1. Give an example of each of the following. No justication is needed.
(a) A closed and bounded subset of C[0, 1] which is not compact.
(b) An unbounded subset of R3 which is not connect
Math 320-2: Lecture Notes
Northwestern University, Winter 2015
Written by Santiago Caez
n
These are lecture notes for Math 320-2, the second quarter of Real Analysis, taught at Northwestern University in the winter of 2015. The book used was the 4th editi
Math 320-2: Lecture Notes
Northwestern University, Winter 2015
Written by Santiago Ca
nez
These are lecture notes for Math 320-2, the second quarter of Real Analysis, taught at Northwestern University in the winter of 2015. The book used was the 4th editi
Math 320-2: Lecture Notes
Northwestern University, Winter 2015
Written by Santiago Caez
n
These are lecture notes for Math 320-2, the second quarter of Real Analysis, taught at Northwestern University in the winter of 2015. The book used was the 4th editi
Notes on Metric Spaces
These notes introduce the concept of a metric space, which will be an essential notion throughout
this course and in others that follow. Some of this material is contained in optional sections of
the book, but I will assume none of
Notes on Metric Spaces
These notes introduce the concept of a metric space, which will be an essential notion throughout
this course and in others that follow. Some of this material is contained in optional sections of
the book, but I will assume none of
Math 320-2: Lecture Notes
Northwestern University, Winter 2015
Written by Santiago Caez
n
These are lecture notes for Math 320-2, the second quarter of Real Analysis, taught at Northwestern University in the winter of 2015. The book used was the 4th editi
Math 320-2: Lecture Notes
Northwestern University, Winter 2015
Written by Santiago Caez
n
These are lecture notes for Math 320-2, the second quarter of Real Analysis, taught at Northwestern University in the winter of 2015. The book used was the 4th editi
Problem Set 5 Solution Set
Anthony Varilly
A
(adapted from Nitin Saksenas L TEXle)
Math 112, Spring 2002
1. End of Chapter 4, Exercise 6b. Let f : R R be a bounded function. Prove that f is
continuous if and only if the graph of f is a closed subset of R2
The Completion of a Metric Space
Let (X, d) be a metric space. The goal of these notes is to construct a complete metric space
which contains X as a subspace and which is the smallest space with respect to these two
properties. The resulting space will be
Math 320-3: Midterm 1 Practice Solutions
Northwestern University, Spring 2015
1. Give an example, with justication, of each of the following.
(a) A limit lim(x,y,z,w)(0,0,0,0) f (x, y, z, w) which does not exist.
(b) A function f : R2 R which is not diere
Math 320-3: Midterm 2 Practice
Northwestern University, Spring 2015
1. Give an example, with justication, of each of the following.
(a) A Jordan region E R2 which contains a non-Jordan region E0 E.
(b) An integrable function on [0, 1] [0, 1] which is disc
Math 320-3: Midterm 1 Practice
Northwestern University, Spring 2015
1. Give an example, with justication, of each of the following.
(a) A limit lim(x,y,z,w)(0,0,0,0) f (x, y, z, w) which does not exist.
(b) A function f : R2 R which is not dierentiable at
Math 320-1: Final Exam
Northwestern, Fall 2013
December 12, 2013
Name:
1. (15 points) For each of the statements below, decide if it is true or false. If true, prove it; if
false, give a counterexample.
(a) For functions f and g on [0, 1],
sup ( f ( x) +
Math 320-1: Final Exam Solutions
Northwestern University, Fall 2013
1. For each of the statements below, decide if it is true or false. If true, prove it; if false, give a
counterexample.
(a) For functions f and g on [0, 1],
sup (f (x) + g (x) = sup f (x)