References texts:
- [M] Ray Meyer, Math 112 course notes
- [P] Dave Perkinson, Math 112 lecture notes
- [H] Richard Hammack, Book of Proof
The contents of [M] and [P] are, of course, most relevant to our proof-based course focused on analysis (as opposed to on discrete math, number theory, or abstract algebra). A more systematic discussion on logic (connectives, quantifiers, truth tables) and proof techniques (direct, by induction, by contradiction, by contraposition) is found in [H].
Homework solutions must be typeset with $\rm\LaTeX$ (see getting started) and submitted on GradeScope (see Moodle page for the entry code). Working together is encouraged, as collaboration is a nontrivial part of learning and of scholarship in general. But, in accordance with the honor principle and basic human decency, you must submit your own write-up with an acknowledgment of collaborators.
One in-class midterm and one in-person cumulative exam during university-scheduled final exam week will be administered.
TA info: Olivia McGough (mcgougho@) is our teaching assistant. She will be holding optional discussion sessions M 5:30 p.m.–7:30 p.m. at Library 389.
Math Help Center/Drop-in Tutoring: SuMTWTh 7:00–9:00 p.m. at Library 204 (schedule).
Individual Tutoring: Reed offers one hour per week of free, one-to-one tutoring. Details here.
Week
Date
Topics Covered
Suggested Reading
1/23
Set theory: basic notations and operations
[
M, pp. 1–9, 14–15]
[
P, pp. 16–19]
[
H, pp. 3–8, 12–15, 18–19]
1/25
More set theory (power sets and equivalence relations deferred to Math 113); some sample proofs involving sets
[
M, pp. 15–16]
[
P, pp. 20–22]
[
H, pp. 8–11]
When presenting an argument, I will attempt to appeal to your common sense instead of making a spectacle of the underlying logic. Those interested in logic (e.g., $P \to Q = \neg Q \to \neg P$) are encouraged to consult [
H, pp. 34–64]. Also, [
H, pp. 113, 128, 138] discusses the logical backbone and presents proof templates of the common proof techniques.
1/27
Irrationality of $\sqrt{2}$
[
H, Proposition on p. 139]
This is meant as a demonstration of "good" writing style (e.g., use of lemmas, "without loss of generality," "for the sake of a contradiction, assume...") and common proof techniques (direct, contrapositive, contradiction). Again, [
H] contains background in logic as well as additional proof examples and style suggestions.
Homework 1 (pdf, tex, solutions) due on Monday, January 30
1/30
Functions, (co)domain, image/range, injective/surjective, composition
[
P, pp. 29–31]
[
H, pp. 225–232, 235–238]
I did not take the abstract approach, as most of the recommended references do, of defining functions as relations, i.e., as subsets of the Cartesian product of the domain and the codomain.
2/1
Inverse function and bijections
[
P, pp. 33–36]
[
H, pp. 238–241]
2/3
Cardinality and Cantor's diagonal argument
[
P, pp. 33–36]
[
H, pp. 238–241, 269–283]
It appears that cadinality is discussed only in [
H, pp. 269–283]. It is worth a read if you are interested in sizes of infinities; the pictorial representations of the various bijections (e.g., on p. 277) are particularly nice.
Homework 2 (pdf, tex, solutions) due on Monday, February 6
2/6
Cantor's diagonal argument and field axioms
[
M, pp. 29–43]
[
P, pp. 41–47]
2/8
Field axioms
[
M, pp. 29–43]
[
P, pp. 41–47]
For consistency, I will henceforth be using the labels (A1)–(A4), (M1)–(M4), (D) from [
P, pp. 41–42] when referring to the field axioms.
2/10
Consequences of field axioms
[
M, pp. 29–43]
[
P, pp. 41–47]
Homework 3 (pdf, tex, solutions) due on Monday, February 13
For consistency, I will henceforth be using the labels (O1)–(O4) from [
P, p. 48] when referring to the order axioms.
[
M] defines an ordered field in terms of the positive cone. This is equivalent to Perkinson's (and our) presentation, but to avoid confusion I will not assign reading from Meyer. As always,
Wikipedia is available for those interested.
2/15
Least upper bounds
[
M, pp. 97–99]
[
P, pp. 52–54]
2/17
least upper bound property and archimedean property of $\mathbf{R}$
[
M, pp. 97–105]
[
P, pp. 52–59]
Homework 4 (pdf, tex, solutions) due on Monday, February 20
2/20
Consequences of the archimedean property
[
M, pp. 97–105]
[
P, pp. 52–59]
There is a lot to unpack in the last few lectures. I recommend carefully going over the relevant sections of [
P, pp. 52–59]. First, make sure you are comfortable with the basic definitions on p. 52. Next, examine the examples given on pp. 52–53. Third, review the least upper bound property (aka completeness) on pp. 53–54. Finally, go over Propositions 1–4 on pp.56–58 (we did similar, but not identical, things in class).
2/22
the field $\mathbf{C}$ of complex numbers
[
M, pp. 83–91]
[
P, pp. 59–72]
2/24
arithmetic and geometry of $\mathbf{C}$
Snow day
Homework 5 (pdf, tex, solutions) due on
Monday Wednesday, March 1
Some notes on what we covered in class.
2/27
Arithmetic and geometry of $\mathbf{C}$
[
M, pp. 83–91]
[
P, pp. 59–72]
If this is your first time seeing complex numbers (in a long while), then a good idea is to step away momentarily from the abstract symbol-pushing that has been happening in class to practice basic algebra using specific complex numbers. For example, try the exercises from
OpenStax.
3/3
Mathematical induction
[
P, pp. 11–15]
[
H, pp. 180–197]
3/10
Sequences and limits ($\epsilon$-$\delta$ definition)
[
M, pp. 125–129]
[
P, pp. 78–82]
3/20
Sequences and limits ($\epsilon$-$\delta$ definition)
[
M, pp. 125–129]
[
P, pp. 78–82]
3/22
More computations of limits; limit laws
[
M, pp. 136–144]
[
P, pp. 83–94]
3/24
Proofs of limit laws
[
M, pp. 136–144]
[
P, pp. 83–94]
3/27
Some special sequences
3/29
Monotone convergence theoren
3/31
Cauchy sequences and Cauchy completeness of $\mathbf{R}$
Homework 6 (pdf, tex, solutions) due on Monday, April 3
4/3
Series: notation and examples, geometric series
4/5
Test for divergence, comparison test
4/7
Comparison test (examples), Cauchy condensation test, $p$-series
Homework 7 (pdf, tex, solutions) due on Monday, April 10
4/10
More examples of comparison/condensation test, the number $e$
4/14
Absolutel convergence, alternating series test, rearrangement
Homework 8 (pdf, tex, solutions) due on Monday, April 17
4/17
Wrapping up series: absolute convergence and Riemann's rearragement theorem
Limits and continuity
[
P, pp. 112–117, 126–132]
Homework 9 (pdf, tex, solutions) due on Monday, April 24
4/24
Power series and radius of convergence
4/28
Defining $\exp(z)$ and Euler's formula