A Model Argument. Eulers Number.
1
Mth 311 Fall 1997 Bent Petersen.
One of the biggest problems for students in advanced calculus is the requirement
to provide proofs. More precisely the problem is guring out what constitutes a
proof. Here I will provide
Homework assignments:
HW assignments (collected):
HWI: 1.1: #5 (Use Example 1.1) and #16; 1.2: #4a (use suggested
problems 1.1#15 and 1.2 #3, but don't turn in the proofs of these) (due
Mon Jan 13). Note the typo on p.1: a square is missing in the very la
3.5 # 7
a. Prove that f (x) = x, x [0, 1] is continuous, i.e. if cfw_xn is a sequence in [0, 1], and
xn x0 in [0, 1] as n , then xn x0 .
2 cases: If x0 = 0, then:
xn x0
1
| xn x0 | = |
| |xn x0 |
xn + x0
x0
By the comparison lemma, the result follows.
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2.4 # 8 If cfw_an is monotone and has a convergent subsequence, then cfw_an converges.
Assume that cfw_an is monotonically increasing (if decreasing, similar argument). It suces to
show that cfw_an is bounded above, by the Monotone Convergence Theorem
Mth 311 October 28, 1997
Mth 311 Limit Inferior and Limit Superior
In this note we will discuss the limit inferior and limit superior of a sequence in a bit more detail than in
the text. Both the limit inferior and the limit superior may be viewed as a re
Solution Sketches
Problems 1 - 5 Mth 311 Fall 1997 Bent Petersen
Here are solution sketches for the rst 5 problems. I have avoided giving you complete
detailed solutions since you should provide your own.
Problem 1.
Suppose the set A has at least two elem
Mth 311 November 7, 1997
Mth 311 The Zero Convergence Test
The simplest series convergence test is the zero Test.
Proposition 1 (Zero Test 1). If
k=1
ak converges then limk ak = 0.
The converse is true for alternating series (Zero Test 2), though false in
Mth 311 November 19, 1997
AbelDedekindDirichlet Theorem
Recall the alternating series test:
Theorem 1 (Alternating Series Test). If
(1) (bn )n1 is monotone decreasing, and
(2) limn bn = 0
then the series
n
(1) bn
n=1
converges.
This result is easily prove
Mth 311 Assignment 1
Due: Oct 13, 1997
Please turn in neat carefully written solutions to the problems. You should try to write good proofs.
Problem 1. Let A be a set. A sequence in A is a function a : N A. If we set ak = a(k) for each
k N we sometimes w
Mth 311 Assignment 2
Due: Oct 17, 1997
Please turn in neat carefully written solutions to the problems. You should try to write good proofs.
You may discuss the problems with anyone for the purpose of obtaining ideas and clarication. You
are expected how
Mth 311 Assignment 3
Due: Oct 29, 1997
Please turn in neat carefully written solutions to the problems. You should try to write good proofs. We
are looking for technical details (check the model argument on the web page). You may discuss the
problems wit
Mth 311 Assignment 4
Due: Nov 21, 1997
Please turn in neat carefully written solutions to the problems. You should try to write good proofs. We are
looking for technical details (check the model argument on the web page). You may discuss the problems wit
Mth 311 Exam F97
Bent Petersen
311f97ex.tex
Instructions: =
If you do not read the
instructions, then how
will you know what to
do? Read them now.
Be sure to write
your name in the
space above.
Name:
December 10, 1997
Time: 110 minutes.
You may use one n
Mth 311 Test 1 F97
Bent Petersen
311f97q1.tex
Instructions: =
If you do not read the
instructions, then how
will you know what to
do? Read them now.
Be sure to write
your name in the
space above.
Name:
November 21, 1997
Time: 50 minutes.
You may use one
Mth 311 Review Problems
November 17, 1997
1
We will have an inclass test Friday November 21. Here is a brief list of some of the topics we
will have discussed before the test:
1. Set theory and cardinality
2. Sequences and completeness of R
Cauchy sequen