Marco Varisco,
marco@math.binghamton.edu
[how to email a professor],
math.binghamton.edu/marco/proofs/
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.
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 on | Problems |
---|---|---|
1 | R 1/29 | Prove propositions 1.7 & 1.10(ii). |
2 | M 2/02 | Prove propositions 1.14, 1.15, & 1.20. |
3 | W 2/04 | Prove propositions 1.16 & 1.19, and read chapter 2. |
4 | R 2/05 | Reread chapter 2, and do projects 2.1 & 2.2. |
5 | M 2/09 | Retake (but do not hand in) quiz #1, and do this. |
6 | R 2/12 | Prove proposition 3.10(ii). |
7 | F 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, n^{2}+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. |
8 | W 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.) |
9 | W 2/25 | Retake (but do not hand in) quiz #2,
and prove that for any natural number k, |
10 | W 3/04 | Find a formula for for any natural number k, and prove that your formula is correct. |
11 | R 3/05 | Prove that for all natural numbers x and y, there exists a natural number n such that nx≥y. |
12 | W 3/11 | Prove theorem 6.8. Prove that any non-empty, bounded above set of integers has a largest element. |
13 | F 3/27 | Prime time! |
14 | M 3/30 | Retake (but do not hand in) quiz #3, and prove propositions 10.6 & 10.12 (you may use proposition 10.11). |
15 | M 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. |
16 | W 4/22 | Retake (but do not hand in) quiz #5, and prove that the sequence x_{n}=(-1)^{n} diverges. |
17 | M 4/27 | Prove that the sequence defined recursively by x_{1}=2 and x_{n+1}=(x_{n}+6)/2 is convergent. |