Binghamton University, SUNYMath 330: Introduction to Higher Math

(aka Number Systems, or simply Proofs)

Section 1, Spring 2009


Marco Varisco, [how to email a professor],
Office: LN-2208A, Office Hours: MW 3:00–4:30 or by appointment.


MWF 10:50–11:50 in SW-325 and R 10:05–11:05 in LN-G335.
N.B.: The Thursday meeting is not a discussion session.

You are expected to attend all class meetings. The maximum number of absences permitted to receive credit for this course is 5 (five). Excessive tardiness may count as absence. [University Bulletin]


Math 222: Calculus II with a grade of C or better. [University Bulletin]


“Careful discussion of the real numbers, the rational numbers and the integers, including a thorough study of induction and recursion. Countable and uncountable sets. The methodology of mathematics: basic logic, the use of quantifiers, equivalence relations, sets and functions. Methods of proof in mathematics. Training in how to discover and write proofs.” [University Bulletin]


The Art of Proof: A Concrete Gateway to Mathematics, Matthias Beck and Ross Geoghegan, 2009.
This textbook is published by Binghamton University and is only available from local bookstores.

Grading & Examinations

When calculating your course grade there is one more rule: if your homework score is an F then your course grade is an F; in this case I will ignore your midterm, project, and final exam scores.

Of course, you are expected to obey the Student Academic Honesty Code.


#Due onProblems
1R 1/29 Prove propositions 1.7 & 1.10(ii).
2M 2/02 Prove propositions 1.14, 1.15, & 1.20.
3W 2/04 Prove propositions 1.16 & 1.19, and read chapter 2.
4R 2/05 Reread chapter 2, and do projects 2.1 & 2.2.
5M 2/09 Retake (but do not hand in) quiz #1, and do this.
6R 2/12 Prove proposition 3.10(ii).
7F 2/13 An integer n is called even if it is divisible by 2, i.e., if there exists an integer k such that n=2k.
Question: Is 0 even? Why or why not?
Prove the following statements:
[A] If m and n are even then m+n and mn are even.
[B] For any natural number n, n2+n is even.
[C] For any natural number n, n is even or n+1 is even.
[D] For any integer n, n is even or n+1 is even.
8W 2/18 Prove propositions 3.17, 3.19, 3.20, & 3.23.
An integer n is called odd if it is not even.
Question: Is 1 odd? Why or why not?
Prove the following statements:
[E] For any integer n, either n is even or n+1 is even, but not both.
[F] If n is an odd integer, then there exists an even integer m such that n=m-1.
(You may use statements [A], [B], [C], & [D] from the previous assignment.)
9W 2/25 Retake (but do not hand in) quiz #2, and prove that for any natural number k,
\sum_{j=1}^k 2^j = 2^{k+1}-2, \sum_{j=1}^k j2^j = (k-1)2^{k+1}+2.
10W 3/04 Find a formula for
\sum_{j=1}^k (2j-1)
for any natural number k, and prove that your formula is correct.
11R 3/05 Prove that for all natural numbers x and y, there exists a natural number n such that nx≥y.
12W 3/11 Prove theorem 6.8.
Prove that any non-empty, bounded above set of integers has a largest element.
13F 3/27 Prime time!
14M 3/30 Retake (but do not hand in) quiz #3, and prove propositions 10.6 & 10.12 (you may use proposition 10.11).
15M 4/20 [1] Prove that the sequence (n-1)/n converges to 1.
[2] Write down explicitly what it means for a sequence to be divergent.
[3] Prove that any convergent sequence of real numbers is bounded above.
[4] Prove that any non-decreasing and bounded above sequence of real numbers is convergent.
16W 4/22 Retake (but do not hand in) quiz #5, and prove that the sequence xn=(-1)n diverges.
17M 4/27 Prove that the sequence defined recursively by x1=2 and xn+1=(xn+6)/2 is convergent.

This syllabus is subject to change. All official announcements and assignments are given in class, and this web page may not be up to date.
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