Week 1
☛ Monday ☚
Introduction. Proof by induction.
☛ Wednesday ☚
More induction.
Homework due Wednesday
- Skim sections 1.1–1.3.
- Read section 1.4 and section 1.5 through page 38.
- Write a solution to exercise 1.5.1 in a form suitable to be handed in
(it will not be collected, though).
☛ Friday ☚
Sets.
Homework due Friday
Week 2
☛ Monday ☚
More sets. Cartesian products.
Homework due Monday
- Relevant reading: Sections 2.1 and 2.2.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H02T (tex file: H02T.tex).
☛ Wednesday ☚
Relations and equivalence relations.
(See the
mathematical writing
handout for a template for proving a given relation is an equivalence
relation.)
Homework due Wednesday
- Relevant reading: Section 2.3.
☛ Friday ☚
Equivalence classes. Modular arithmetic. Functions.
Homework due Friday
- Relevant reading: Section 2.4.
- Turn in:
H02F
(tex file: H02F.tex).
Week 3
☛ Monday ☚
Functions.
Homework due Monday
- Reading: Finish reading Section 2.4.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H03T (tex file: H03T.tex.)
☛ Wednesday ☚
Injectivity, surjectivity, bijectivity. Functions acting on sets.
Homework due Wednesday
- Reading: Start reading Section 2.5 in preparation for Friday.
☛ Friday ☚
Compositions and inverse functions.
Homework due Friday
- Reading: Sections 2.5 and 2.6.
- Turn in:
H03F
(tex file: H03F.tex).
Week 4
☛ Monday ☚
Field axioms. Modular arithmetic.
Homework due Monday
- Reading: Read the first page of our handout on the
complete ordered field axioms.
Finish reading Section 2.5, and start Section 2.6.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H04T (tex file: H04T.tex).
☛ Wednesday ☚
More modular arithmetic. Begin field properties.
Homework due Wednesday
- Reading: Sections 2.5 and 2.6.
☛ Friday ☚
More field properties.
Homework due Friday
- Reading: Finish reading Section 2.6.
- Turn in:
H04F
(tex file: H04F.tex).
Week 5
☛ Monday ☚
More implications of field axioms. Order axioms.
Homework due Monday
- Reading: Finish reading section 2.6.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H05T (tex file: H05T.tex).
☛ Wednesday ☚
Absolute value. Completeness.
Homework due Wednesday
☛ Friday ☚
Midterm.
Homework due Friday
- Reading: None
- Turn in: No homework this Friday.
Week 6
☛ Monday ☚
Sups, infs, maxs, and mins.
Worksheet.
Homework due Monday
- Reading: Last Friday's lecture notes and Section 2.7.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H06T (tex file: H06T.tex).
☛ Wednesday ☚
Complex numbers.
Homework due Wednesday
☛ Friday ☚
Complex numbers.
Homework due Friday
- Reading: Sections 3.1 and 3.2.
- Turn in:
H06F
(tex file: H06F.tex).
Week 7
☛ Monday ☚
Geometry of complex numbers.
Homework due Monday
- Reading: Sections 3.2 and 3.3 and lecture notes.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H07T (tex file: H07T.tex).
☛ Wednesday ☚
Topology.
Homework due Wednesday
- Reading: Sections 3.3 and 3.4.
☛ Friday ☚
Sequences.
Homework due Friday
- Reading: Start reading chapter 8.
- Turn in:
H07F
(tex file: H07F.tex).
Week 8
☛ Monday ☚
Limits of sequences.
Homework due Monday
- Reading: Sections 8.1 and 8.2.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H08T (tex file: H08T.tex).
☛ Wednesday ☚
Limit theorems.
Homework due Wednesday
- Reading: Today's lecture notes. (The ideas can also be found in
Chapter 8.)
☛ Friday ☚
Proof of the limit theorem.
Homework due Friday
- Reading: Today's lecture notes. (The ideas can also be found in
Chapter 8.)
- Turn in:
H08F
(tex file: H08F.tex).
Week 9
☛ Monday ☚
Convergent sequences are bounded. The squeeze theorem. Taking limits preserves inequalities.
Homework due Monday
- Reading: Chapter 8 (especially Section 8.4).
- Turn in by 2 pm on Tuesday in the folder outside my office:
H09T (tex file: H09T.tex).
☛ Wednesday ☚
Monotone convergence theorem.
Homework due Wednesday
☛ Friday ☚
Subsequences, infinite limits, Cauchy sequences.
Homework due Friday
- Reading: Sections 8.3, 8.7, and 8.8.
- Turn in:
H09F
(tex file: H09F.tex).
Week 10
☛ Monday ☚
Series. Geometric series.
Homework due Monday
- Reading: Section 9.1.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H10T (tex file: H10T.tex).
☛ Wednesday ☚
A criterion for divergence. The harmonic series. The comparison theorem.
Homework due Wednesday
☛ Friday ☚
Series tests. See our
series tests handout.
Homework due Friday
Week 11
☛ Monday ☚
Series tests. See our
series tests handout.
Homework due Monday
- Reading: Section 9.2.
- Turn in by 2 pm on Tuesday in the folder outside my office:
No Tuesday homework this week.
☛ Wednesday ☚
Limits of functions.
Homework due Wednesday
- Reading: Sections 4.1–4.3.
☛ Friday ☚
Limit theorems, continuity and derivatives.
Homework due Friday
- Reading: Sections 5.1, 6.1, and 6.2.
- Turn in:
H11F
(tex file: H11F.tex).
Week 12
☛ Monday ☚
Power series.
Homework due Monday
- Reading: Section 9.3.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H12T (tex file: H12T.tex). For problem 3, you
may use our limit theorems or give an epsilon-delta proof.
☛ Wednesday ☚
Power series.
Homework due Wednesday
☛ Friday ☚
Two main theorems for power series.
Homework due Friday
- Reading: Section 9.3 and 9.4.
- Turn in:
H12F
(tex file: H12F.tex).
Week 13
☛ Monday ☚
Taylor series.
Homework due Monday
- Reading: Sections 6.5 and 9.7.
- Turn in by 2 pm on Tuesday in the folder outside my office:
H13T (tex file: H13T.tex).
☛ Wednesday ☚
Taylor series.
Homework due Wednesday
- Reading: Sections 6.5 and 9.7.
☛ Friday ☚
\(\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}\). Review for final.
Homework due Friday
- Reading: Review for final handout.
- Turn in: No homework due this Friday.