Comments

• Errata to the older edition of our textbook.
• Books on real analysis and on math foundations.
• Topics covered in this course
• Office Hours: M 3-5, W 4-5, F 3-5.

week 1

[20 aug]

Lecture 0: Organizational Meeting [see Topics].
Lecture 1: A. Propositional logic.
Lecture 2: B. Set theory.

week 2

[27 aug]

Lecture 3: C. Predicate logic. D. Functions and relations.
Lecture 4: E. Metric spaces. F. On proofs.
Lecture 5: G. What are real numbers?
Homework 1: Textbook 1.2: 1, 3, 8, 11, 12.

week 3

[3 sep]

Lecture 6: G. Real nrs: Archimedean property. [done as Lecture 7]
Lecture 7: G. Real nrs: Q is dense in R. [done as Lecture 8]
Homework 2: Textbook 1.3: 1, 2ab, 3, 5, 7. 1.4: 4, 5, 6a.
Also prove: ∀ x ∈ R   ∃ n ∈ N     x < n ≤ x + 1.
All due next Monday, 17 Sep.

week 4

[10 sep]

Lecture 8: H. Cardinal numbers. [done as Lecture 6]
Lecture 9: Sequences and series: paradoxes.
Lecture 10: Sequences: algebraic properties.
Homework 3: Problems 2.2: 1, 6, 7, 8. 2.3: 1, 2a, 3, 6, 8 (read 9, 11). [due 21 sep]

week 5

[17 sep]

Lecture 11: Momotone Conv'ce Thm.
Lecture 12: Examples of interesting series. Bolzano-Weierstrass Theorem.
Lecture 13: Cauchy Criterion.
Homework 4: Problems 2.4: 2a, 4*, 5 (read 6 for some info). 2.5: 1, 3*. 2.6: 1, 2, 4* (may try for yourself 5 and 6).
Submit only problems marked by star. Also do but do not submit: 1.4: 11,12 (read 13). 1.5: 4,9.

week 6

[24 sep]

Lecture 14: Series: Algebra, Cauchy and Absolute Conv'ce.
Lecture 15: Series: Rearangements, Cauchy product.
Lecture 16: Preludium to topology: the Cantor set.
Homework 5: Problems 2.7: 4, 5, 9.

week 7

[1 Oct]

Lecture 17: What is topology.
Lecture 18: Open and closed sets, accumulation points.
Lecture 19: More on closed sets.
Homework 6: Problems 3.2: 1, 2, 3, 7, 9*, 12*.
Do all listed, submit only the problems that are marked by a star "*".

week 8

[8 Oct]

Lecture 20: Compact sets.
Lecture 21: More on compact sets. Heine-Borel Thm.
Lecture 22: Disconnected sets and other topological addenda.
Homework 7: Problems 3.3: 1, 2, 4, 5, 7, 9*.

week 9

[15 Oct]

Lecture 23: What are functions (intro).
Lecture 24: Limits of functions (4.2).
Lecture 25: TEST 1.

week 10

[22 Oct]

Lecture 26: Continuity (4.3).
Lecture 27: Continuity and compact sets (4.4)
Lecture 28: Bolzano Thm and Inermediate Value Thrm (4.5).
Homework 8: Problems 4.2: 1a, 6, 9*. 4.3: 2, 3, 5*, 11. 4.4: 2, 3, 4*, 8, 9.
Submit only the problem marked by a star "*".

week 11

[29 Oct]

Lecture 29: Calculus 1: Derivative (5.2).
Lecture 30: Calculus 2: More theorems (5.2-3)
Lecture 31: Calculus 2: More theorems (Darboux thm) (5.3).
Homework 9: Problems 5.2: 2, 4. 5.3: 2*, 5, 7.
Submit only the problem marked by a star "*".

week 12

[5 Nov]

Lecture 32: Sequences of functions: uniform convergence (6.2).
Lecture 33: Limits and differentiability (6.3)
Lecture 34: Series of functions (6.4). Nowere differentiable continuous monster function.
[See Cantor function]
Homework 10: Problems 6.2: 1--5, 11, 13. 6.3: 1, 2, 5. 6.4: 1, 2, 6, 7*. 6.5: 1a, 2.

week 13

[12 Nov]

Lecture 35: Power series (6.5).
Lecture 36: Integrals (a la Riemann-Darboux) (7.2)
Lecture 37: Integrability (7.3-4).
Homework 10: Problems 7.2: 2, 4. 7.3: 4* (guess the value, choose a smart partition). 7.4: 1, 4*, 5b.

week 14

[19 Nov]

Lecture 38: Integrals and algebra (7.4)

BREAK

week 15

[26 Nov]

Lecture 39: Integrability.
Lecture 40: FTC (7.5).
Lecture 41: Lebesgue Theorem (7.6).
The Last Homework: Problems 7.5: 1, 2, 7, 8, 9. 7.6: 1 (none to turn in).

week 15

[3 Dec]

Review