Math 325, 2nd Semester, '08-'09

Math 325-001, Elementary Analysis is offered by the Department of Mathematics of the University of Nebraska-Lincoln. The class meets 12:30-1:20 MWF in Oldfather 208 and is taught by Allan Donsig.

Announcements

A 15 page handout covering mathematical language and proof techniques is available on blackboard.

Exams

Information about exams, and their solutions, will be posted here.

First Midterm

The first midterm test was from 6 to 8 pm on Monday, February 23 in Oldfather 208, our usual classroom. The test covers all of the material we've covered so far, up to Corollary 10.5, plus Definition 10.8 and Lemma 10.9. This includes the background chapter, although the emphasis will be on the material from the textbook, not the material from the background chapter.

You will be asked to prove one of the four following results:

The existence of inf S (Corollary 4.5 in the book)
Sequences with nonzero limit are bounded away from zero (Example 6 in Section 8)
`Products of Limits' Theorem (Theorem 9.4)
'Limits of Reciprocals and Infinity' Theorem (Theorem 9.10)

Here is a list of review exercises from the textbook: 3.6, 4.9, 4.10, 4.15, 5.4, 8.1(c)(d), 8.6, 9.3, 9.4, 9.10, 10.2. You are responsible for all the material covered in the class; there may be problems on the exam that are not similar to the review exercises. (That is, doing review exercises is not a substitute for studying.)

Second Midterm

The second midterm test was from 6 to 8 pm on Wednesday, April 8 in Oldfather 208, our usual classroom. The test covers all of the material we've covered so far, up to the end of Section 15. That is, everything in the book except Sections 6 and 13.

You will be asked to prove one of the three following results:

If limsup s_n equals limingf/i> s_n, then (s_n) converges to this common value (Theorem 10.7(ii)).
Every sequence has a monotonic subsequence (Theorem 11.3)
The convergence part of the Root Test (Theorem 14.9(i))

Here is a list of review exercises from the textbook: 11.4, 11.5, 11.7, 11.10, 12.2, 12.7, 12.8, 12.10, 12.14, 14.3, 14.9, 14.10, 14.12, 15.6, 15.7 You are responsible for all the material covered in the class; there may be problems on the exam that are not similar to the review exercises. (That is, doing review exercises is not a substitute for studying.)
Final Exam

The final exam will be from 3:30 to 5:30 on Wednesday, May 6 in Oldfather 208, our usual classroom. The exam is comprehensive, with the same format as the two midterm tests. The definitions and statements of theorems portion will have a higher emphasis on the material since the second midterm. We have covered everything in the textbook, up the end of Section 24, except Sections 6, 13, 16, 21, and 22.
You will be asked to prove one of the three following results:

the Extreme Value Theorem (Theorem 18.1),
a continuous function on a closed interval [a, b is uniformly continous (Theorem 19.2), and
The uniform limit of continuous functions is continuous (Theorem 24.3).
Here is a list of review exercises from the textbook for material that has not been covered on assignments: 23.5, 23.7, 24.3, 24.5, 24.7, 24.11, 24.13. You are responsible for all the material covered in the class. (That is, doing review exercises is not a substitute for studying.)

Assignments
Assignments will be posted here, in pdf format. The TeX code for the assignments is posted somewhere else. Note that, in addition to the file for the assignment, helpfully called assignNN.tex, you will also need the file newtest.cls, which specifies the macros used in the assignment file. Further, you'll need to be using LaTeX2e (almost certainly, you are) and have access to the amsmath package (which I think is part of the default distribution). Solutions are posted on Blackboard, under "Assignments".

Assignment 0 Now Due on Friday, Jan 16th
Assignment 1
Assignment 2
Assignment 3
Assignment 4
Assignment 5
Assignment 6
Assignment 7
Assignment 8
Assignment 9
Assignment 10
Assignment 11

Quizzes

Here is a list of all the quiz questions. Since the answers are clearly stated in the lectures and the textbook, I don't give them here.

1 Define each of the following: a) $A \subset B$, b) $\varnothing$, c) $A \times B$, and d) $f^{-1}(D)$ where $f : A \to B$ and $D \subset B$. (The notation makes more sense in LaTeX.)
2 State the Principle of Mathematical Induction.
3 In the construction of Z from N, define completely the set we use for Z.
4 Define: the limit of the sequence (a_n) is L.
5 State the theorems for products and sums of limits.
6 Define lim_{n to +∞} s_n = +∞ and lim_{n to +∞} s_n = +∞.
7 Define limsup_{n to +∞} s_n.
8 State the Comparision Test and the Root Test.
9 State the Integral and Alternating Series Tests.
10 Define the intermediate value property and state the Intermediate Value Theorem.
11 Define: a function f : D to R is uniformly continuous on S,a subset of D.

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