1. Introductory Real Analysis. A.N. Kolmogorov & S.V. Fomin. MAIN REFERENCE for the first third of the course.

2. Curso de Análise, vol 1. Elon Lages Lima MAIN REFERENCE for the second third of the course.

3. Principles of Mathematical Analysis. Walter Rudin MAIN REFERENCE for the last third of the course.

4. Real Analysis for the Undergraduate. With an invitation to Functional Analysis Matthew A Pons.

5. Real Analysis via Sequences and Series. Little, Teo and van Brunt

6. Basic Real Analysis. Houshang H. Sohrab

7. Real Analysis. Foundations and Functions of One Variable. Miklos Laczkovich, Vera T. Sos

8. Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch.

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1. Before the First Session. Read the the Section To the Student, page xi to xii in Real Analysis for the Undergraduate. With an invitation to Functional Analysis Matthew A Pons and read the introduction from Curso de Análise, vol 1. Elon Lages Lima.

2. August the 2th, 2016. Introduction of the course. Evaluation system. Bibliography. Scope lecture on Mathematical Analysis and Strategic Guidelines for Mathematical Problem Solving. Quick review on previous concepts: Sets, examples of finite and infinite sets: sets of numbers , sets of functions, sets of sets. Operations between sets: union, intersection, difference, relative complement, cartesian product, power set of a set. Relations, definition and basic examples, comments on equivalence relations and order relations. Functions, definition, injectivity, surjectivity, bijectivity. Capital example: The indicator function of a set. Contents from Curso de Análise, vol 1. Elon Lages Lima, pages 1-13.

3. August the 4th, 2016. Lecture in English. First Principle of Mathematical Induction (PMI). Well-Ordering Principle of the Natural Numbers (WOP). Several examples of different nature. Comments on the technicalities between induction approach vs analytical approach. Comments on the versatile nature of the mathematical induction. Contents from Curso de Análise, vol 1. Elon Lages Lima, pages 39-42.

4. August the 9th, 2016. Second Principle of Mathematical Induction or Strong Principle of Mathematical Induction. Examples of application. Comments and examples on definitions by recurrence, Exemplo 2, 3, 5, pg 41-42 from Curso de Análise, vol 1. Elon Lages Lima. Cardinality, definition. Proof of equivalence in the finite case. Basic examples for the infinite case related to the set of Natural Numbers. Homework 1 assignment, Due August 23rd 2015.

5. August the 10th, 2016. Debate Session 1. Discussion of problems 12, 15, 22 and 24. Discussion of the equivalence between the well ordering principle of the natural numbers and the second principle of mathematical induction. Discussion on the principle of mathematical induction, starting from a natural number different from 1.

6. August the 11th, 2016. Lecture in English. Cardinality, second part. Examples of bijections between the Natural Numbers and the Natural Numbers multiplied by a finite set, between the Natural Numbers and the Natural Numbers multiplied by itself, between the Natural Numbers and the Rational Numbers. The sequences with values on 0,1 and the power set of the Natural Numbers. The Cantor's Theorem. Definition of order between cardinals, by means of injections. The Continuum Hypothesis and remarks about its relation with the Zermelo-Frank systems of set axioms, Godel's Incompleteness Theorem, Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch. Definition of countable and uncountable sets. Construction of a countable set Y within an uncountable set X and stability of the cardinal ater its removal X - Y. Equipotence between the power set of the Natural Numbers and the interval (01,) as well as with any non-degenerate interval and the whole line of the Real Numbers. Countable union of countable sets. The "modulo Q" subset of the Real Numbers "R". The Cantor-Bernstein-Schroeder Theorem (without proof) for deriving equality of cardinals from two independent inequalities and associated comments, Proofs and Fundamentals, a first course in abstract mathematics. Ethan D. Bloch.

7. August the 16th, 2016. Suspended due to Lecturer's appointment with the Colombian Migration Service.

8. August the 17th, 2016. Debate Session 2. Discussion of several philosophical aspects of mathematics.

9. August the 18th, 2016. Lecture in English. The real numbers. Distance, open balls, closed balls, pgs 72, 73 Curso de Análise, vol 1. Elon Lages Lima. The Archimedean property, Corollary 1.3.6 Real Analysis for the Undergraduate. With an invitation to Functional Analysis Matthew A Pons. Density of the rationals in the reals Theorem 4 pg 84 , Curso de Análise, vol 1. Elon Lages Lima. Definition of Dense and Separaale sets, pg 83 Curso de Análise, vol 1. Elon Lages Lima. Field axioms of the real numbers pgs 61-63 Curso de Análise, vol 1. Elon Lages Lima. Definition of upper and lower bounds of a set pg 74 and, Example 13, pg 78 Curso de Análise, vol 1. Elon Lages Lima. Definition of the supremum, pg 76 Curso de Análise, vol 1. Elon Lages Lima.

10. August the 19th, 2016. Recovery class. The Axiom of the Supremum, discussion of Example 13, pg 78 Curso de Análise, vol 1. Elon Lages Lima. Lemma 1.2.10 Real Analysis for the Undergraduate. With an invitation to Functional Analysis Matthew A Pons. Comments on the Exercises 1.3.7, 1.3.8, 1.3.9 and 1.3.10 Real Analysis for the Undergraduate. With an invitation to Functional Analysis Matthew A Pons. Sequences, definitions: general sequence, monotone (increasing & decreasing), bounded (lower & upper) and periodic pgs 100 -102; Examples 1, 2 and 3 pg 103 Curso de Análise, vol 1. Elon Lages Lima. Definition of subsequence: Definition 2.2.8 Real Analysis for the Undergraduate. With an invitation to Functional Analysis Matthew A Pons. Intuitive discussion of the limit of a sequence, first paragraph of pg 107 Curso de Análise, vol 1. Elon Lages Lima. Reading for students of examples 1 through 11 pgs 103-106.

11. August the 23th, 2016. Limit of a sequence pg 107, Theorem 1, Theorem 2, Theorem 3 and Theorem 4, pgs 109-111, Curso de Análise, vol 1. Elon Lages Lima. Reading for students of examples 1 through 11 pgs 111-114, proof Corollary of pg 110.

12. August the 24th, 2016. Debate Session 2. Discussion of Problems 25, 31 and 41 Homework 1. Discussion of finite sets characterization according to Cantor's definition. Discussion of problems 2, 3 and 4 pg 153 and discussion of Theorem 3 pg 110 Curso de Análise, vol 1. Elon Lages Lima.

13. August the 25th, 2016. Lecture in English. Theorem 5, Theorem 6 and deep discussion on the necessity of the involved hypotheses. Theorem 7, Corollary 1, Corollary 2 pgs 118, 119, Theorem 9. First contact with a sequence of suprema and infima, Reading for students of pg 122 in preparation for next class Curso de Análise, vol 1. Elon Lages Lima.

14. August the 30th, 2016. Theorem 8. Definition of Cluster Points, pg 121, Definition of limSup, and limInf for bounded sequences, pg 122, Theorem 10, Curso de Análise, vol 1. Elon Lages Lima.

15. August the 21st, 2016. Debate Session 3. Discussion of the Supremum as cluster point. further conceptual discussion of limSup, limInf, Discussion of examples 8 and 9 pgs 104-105, Curso de Análise, vol 1. Elon Lages Lima.

16. September the 1st, 2016. Lecture in English. Corollary 1 and 2 pg 123, Theorem 11. Definition of Cauchy Sequences pg 126. Theorem 12, Lemma 1 pg 126, Lemma 2 pg 127. Reading for students of Corollary 1 pg 124, Corollary 2 pg 125 Curso de Análise, vol 1. Elon Lages Lima, also, in preparation for the next class, understand the Statement of Theorem 13 and think about a possible strategy. Homework 2 assignment, Due September 17th 2015.

17. September the 6th, 2016. Theorem 13, Example 19. Definition of Limits to Infinity, pg 130, examples, Theorem 14, 1 and 3. Reading Assignment Theorem 14, 2 and 4, Curso de Análise, vol 1. Elon Lages Lima.

18. August the 7th, 2016. Debate Session 4. Discussion of the Cardinality: countable union of uncountable sets is still uncountable. Discussion of recurrence sequences and convergence.

19. September the 8th, 2016. Lecture in English. Definition of Series pg 134 and Series Convergence, examples, The Integral Criterion. Theorem 15, discussion on its converse. Examples 23, 24, 26 and 27. Theorem 16, Reading Assignment Corollary pg 137, Example 29. Theorem 17, Theorem 18, Reading Assignment Corollary 1, Corollary 2 pg 140 Curso de Análise, vol 1. Elon Lages Lima. First Test September 19th, from 6:00 am to 14:00 pm, Room 43-302.

20. September the 13th, 2016. The Course starts being fully lectured in English. Theorem 19, Corollary 1, Corollary 2. Theorem 20. Multiple examples, Curso de Análise, vol 1. Elon Lages Lima.

21. August the 14th, 2016. Debate Session 5. Discussion of problems 1.1, 1.5 and 1.15

22. September the 15th, 2016. Conditional and Absolute Convergence of Series. Example 30 pgs 138-139, Example 38 pgs 147-148. Definition of positive and negative parts of the sequence and the series pg 147. Discussion on the convergence and divergence of the positive and negative part of the series, depending on absolute and conditional convergence. Definition of Permutation for a Sequence. Theorem 22. Curso de Análise, vol 1. Elon Lages Lima.

23. September the 20th, 2016. Theorem 22, Theorem 24 complete. Theorem 23 to be completed in the next lecture. Curso de Análise, vol 1. Elon Lages Lima.

24. September the 27th, 2016. Completion of Theorem 23. Curso de Análise, vol 1. Elon Lages Lima.

25. September the 29th, 2016. New Chapter Topology. Definition of interior point and open sets, pg 163, Examples 1, 2, 3, 4, 5, 6. Theorem 1, Corollary pg 166, Examples 7 and 9. Intermediate Proposition on countable unions of nested intervals. Curso de Análise, vol 1. Elon Lages Lima. Homework 3 assignment, Due October 27th 2015.

26. October the 4th, 2016. Definition of Cluster Point pg 169 and Closed Sets pg 170, several examples. Theorem 3 , Corollary 2 pg 170 (Corollary 1 HW), Examples 10 and 11, Theorem 4, Corollary pg 171, Curso de Análise, vol 1. Elon Lages Lima.

27. October the 6th, 2016. Theorem 5, Examples 12, 13. Theorem 6. Definition of limits points, pg 175. Examples, Theorem 7. Curso de Análise, vol 1. Elon Lages Lima.

28. October the 11th, 2016. Definition of Isolated Points pg 177, Examples. Theorem 8, Corollary 1 anc Corollary 2 pg 177, Definition of "right limit point". Definition of Topology, horizons. Curso de Análise, vol 1. Elon Lages Lima.

29. October the 13th, 2016. Theorem 9 and Technical Lemma, pg 179, Corollary 1. Contrast against Q, the rational numbers. Compact Sets. Definition of Open Coverings, pg 180, Definition of Compact Sets , Examples. Curso de Análise, vol 1. Elon Lages Lima.

30. October the 18th, 2016. Theorem 10, Extension of Theorem 10, Final Form of the Borel-Lebesgue Theorem, pg 181, Example 24. Beginning of Theorem 11. Curso de Análise, vol 1. Elon Lages Lima.

31. October the 20th, 2016. Theorem 11, conclusion. Bolzano-Weirstrass corollary, pg 183, Theorem 12. A taste of Measure Theory: Proposition 1 pg 184. Curso de Análise, vol 1. Elon Lages Lima.

32. October the 24th, 2016. A taste of Measure Theory: Proposition 2 pg 185, Proposition 3 pg 186. Discussion on bounding the tail of a series for an open set. New Chapter, LIMITS, Definition and discussion of limit as a cluster point. Introduction of the "punctured ball" neighborhood. Theorem 1 on uniquenes of the limit. Curso de Análise, vol 1. Elon Lages Lima.

33. October the 26th, 2016. Theorem 4, Theorem 5, Corollary 1, Corollary 2 pg 199. Theorem 6 and comments on Corollary 1, pg 199 and Corollary 2 pg 200. Comments on Theorem 7. Curso de Análise, vol 1. Elon Lages Lima.

34. October the 31st, 2016. Theorem 9, definition of lateral limits, pg 205, examples. Theorem 11 (Theorem 10 omitted) , Theorem 12, definition of limits to infinity, pgs 208, 209. Curso de Análise, vol 1. Elon Lages Lima.

35. November the 2nd, 2016. Example of limits to infinity. Discussion of properties 1, 3, 4, 6, 8 and 9, pg 210. Definition of cluster points for the continuous, pg 213. Theorem 13, Corollaries 1, 2 and 3, pg 214 omitted and left as reading material. Definition of liminf and limsup in the continuous case, pg 215. Curso de Análise, vol 1. Elon Lages Lima.

36. November the 8th, 2016. Definition of Cluster points set CP(f, a) for a function f at a given point a. Corollary 1, 2, 3 pg 214, Examples of cluster sets satisfying certain cardinality and boundedness requirements. Theorem 14, Theorem 15, Corollary pg 217. Curso de Análise, vol 1. Elon Lages Lima.

37. November the 9th, 2016. New Chapter, Continuity. Definition of continuity and discussion of examples 1, 2, 3 and 4 pg 223. Theorem 1, Theorem 2. Curso de Análise, vol 1. Elon Lages Lima.

38. November the 15th, 2016. Theorem 3, Corollary pg 225, discussion of example on pg 225, Theorem 4, discussion of Corollaries 1, 2 pg 226, Theorem 5 and Theorem 6 . Curso de Análise, vol 1. Elon Lages Lima.

39. November the 16th, 2016. Example 2 pg 227, Corollary pg 228, comments on Theorem 7, discussion of examples. Corollary pg 229, comments on Theorem 8, example 5 pg 229. Discontinuities, definition of first type and second type of discontinuities, pg 23, discussion of examples. Curso de Análise, vol 1. Elon Lages Lima.

40. November the 21st, 2016. Cancelled due to lack of quorum.

41. November the 23th, 2016. Cancelled due to lack of quorum.

42. November the 28th, 2016. Cancelled due to lecturer's circumstances.

43. November the 30th, 2016. Theorem 9, Theorem 10, Corollary pg 232, Corollary pg 234. Curso de Análise, vol 1. Elon Lages Lima.

44. December the 5th, 2016. Cancelled due to lack of quorum.

45. December the 7th, 2016. Theorem 11, Theorem 12, Corollary 1 and Corollary 2 pg 235. Curso de Análise, vol 1. Elon Lages Lima.

46. December the 12th, 2016. Theorem 13, Theorem 14, Corollary pg 239, Theorem 15. Curso de Análise, vol 1. Elon Lages Lima.

47. December the 14th, 2016. Final lecture of the course. Uniform Continuity. Definition pg 240. Examples 21, 22, 23. Definition of Lipschitz functions. Theorem 17, Theorem 16, Corollary pg 243. Example 24, Relationship between continuously differentiable functions and uniform continuity using the Mean Value Theorem, Example 25. Theorem 18, Corollary pg 245. Curso de Análise, vol 1. Elon Lages Lima.