Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
Review Exercises
Juris Stepr ns
a
MATH 2001
Determine whether the following statements are true or false.
Provide justication in either case.
If f is continuous on (a, b ] and [b , c
Math 2001: Real analysis
Class test, Oct 25, 2016
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You have 180 minutes to solve eleven problems.
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The maximal score for this exam is 50. Scores for individual problems add
Math 2001: Real analysis
Class test, Oct 25, 2016
Name:
Student ID:
New password:
(leave blank if unchanged)
You have 180 minutes to solve eleven problems.
Show your complete work.
The maximal score for this exam is 50. Scores for individual problems a
EXAM I SOLUTIONS
Part One
(1) Find an open cover of the set [0, 1] Q with no finite subcover. Justify your
answer.
Solution: Let be any irrational number in [0, 1], e.g., = 2/2. Then
cfw_(1, 1/n) : n N (, 2) is an open cover of [0, 1] Q. Suppose
there was
3.1: 1-7, 13, 14, 22-24.
3.2: 1-10, 13*,14*.
3.3: 1-4 (until you get bored), 6, 7, 8, 10, 11, 13, 14, 15, 16, 17 (solutions to the last two problems
were essentially sketched in class).
3.4: 1-6 (until you get bored), 7-11, 15*, 20, 21, 22, 25.
3.5: 1-3 (
Assignment #3
MATH 2001
#1
L N
By limit definition there exists
Similarly there exists
MN
such that for all
n N if n L|anb|< E for all > 0
n N , if n M
such that for all
|c nb|< E
for all
>0
Pick N=max ( M , L ) n N
If
bn <0 then |bn|<|an|bn b|<|anb|< for
Solution to assigned exercises due December 8,
2010.
(1) Prove that f is continuous at a if and only if limn f (an ) = f (a) for every sequence cfw_an
n=0
such that limn an = a.
Solution 1. If f is continuous at a and limn an = a then to see that limn f
Solutions to exercises due November 24, 2010.
(1) Prove that if an converges then an = a0 + an . This is not as easy as it might
n=0
n=0
n=1
seem, because your proof must start from the denition of the convergence of an .
n=0
Solution 1. Since an = limk
n
Solutions to exercises due November 3, 2010.
(1) Describe a function that is not continuous on the interval [0, 1] but, nevertheless, still satises
the intermediate value property.
Solution 1. Let P be dened by
P (x) =
sin(1/x) if x (0, 1]
0
if x = 0
(Thi
Solutions to exercises due October 27, 2010.
(1) Let f be the function dened by
sin(x)/4
if x /2
2
f (x) = ax + bx + c if /2 < x <
sin(x)
if x
Find values of the parameters a, b and c such that f is dierentiable for all x.
d
Solution 1. It has been sho
Solutions to exercises due October 20, 2010.
(1) By a disk in the plane will be meant a circle together with its interior. Using the nested interval
principle prove that a nested sequence of disks whose radii go to zero has a unique intersection
point. In
Solutions to exercises due October 6.
Question 1. Let T be a right angled, isosceles triangle whose two equal sides have length one. Use the
method of exhaustion to demonstrate that this triangle has area 1/2.
Solution 1. Approximate the area of the trian
Solutions to exercises due September 24.
Question 1. Let (w) (x) be a solution to the heat problem discussed in class. Given that (1) =
(1) = 0 prove that it is not possible for (x) and (x) to be strictly positive for all x such that
1 < x < 1. You may u
Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
September 29, 2010
Juris Stepr ns
a
MATH 2001
Taylor series
If the following hold:
f (x ) = a0 + a1 (x a) + a2 (x a)2 + a3 (x a)3 + . . .
term by term dierentiation is justied in othe
Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
September 23, 2010
Juris Stepr ns
a
MATH 2001
Definition of convergence
Definition
A series converges if there is a target value T such that for any
L < T and any M > T , all of the p
Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
September 21, 2010
Juris Stepr ns
a
MATH 2001
Archimedes and the method of exhaustion
Juris Stepr ns
a
MATH 2001
Archimedes and the method of exhaustion
At each stage n the number of
Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
September 17, 2010
Juris Stepr ns
a
MATH 2001
Solutions to the heat equation
Since and depend on dierent variables it must be that
(w )/ (w ) and (x )/ (x ) are both constant and, hen
Introduction to Real Analysis MATH 2001
Juris Steprns
a
York University
September 13, 2010
Juris Stepr ns
a
MATH 2001
Instructor
Instructor: Juris Steprns
a
My oce is N530 in the Ross building.
Oce hours are Mondays from 4:00 to 5:00 and Wednesdays
from 1
SEQUENCE PROOFS-EXAMPLES
JACOB MILLER
This a collection of things that might help you as you attempt to navigate the scary world
of proving sequences using their formal defintion! This worksheet includes three things:
Resources (i.e. things you should rem