Math 315 FALL 2004, List of lectures

Monday, September 27: Peano axioms for natural numbers; principle of mathematical induction. Section 1.1.

Tuesday, September 28: Integer and rational numbers. Square root of 2 is irrational. Algebraic numbers and Theorem on rational solutions of algebraic equations. Section 1.2.

Wednesday, September 29: Axioms of ordered field and consequences. Section 1.3.

Friday, October 1: Absolute values. Completeness axiom. Sections 1.3, 1.4.

Monday, October 4: Completeness axiom fails for rational numbers. Section 1.4.

Tuesday, October 5: Consequences of the completeness axiom. Infinite supremum and infinum. Real numbers from rationals. Sections 1.4, 1.5, 1.6.

Wednesday, October 6: Sequences, definition of limit, limit is unique if exists. Section 2.7. New homework!

Friday, October 8: Examples of formal proofs for limit calculations. Section 2.8.

Monday, October 11: Limit of square root sequence; any convergent sequence is bounded; multiplication by a scalar. Sections 2.8, 2.9. Today we had first quiz. Here you can find solutions (pdf file).

Tuesday, October 12: Limit of the sum of convergent sequences; limit of the product of convergent sequences. Section 2.9.

Wednesday, October 13: Limit of a ratio; standard examples of limits. Section 2.9. New homework!

Friday, October 15: Infinite limits. Monotone sequences. Sections 2.9 and 2.10.

Monday, October 18: Monotone sequences have limits. Upper limit (lim sup) and lower limit (lim inf). Section 2.10.

Tuesday, October 19: lim sup=lim inf is the same as existense of lim. Cauchy sequences. Convergent sequences = Cauchy sequences. Section 2.10. REMINDER: We will have midterm on Friday; review is on Wedneday (tomorrow).

Wednesday, October 20: Review before midterm. New homework!

Friday, October 22: We had midterm today. Here are the solutions.

Monday, October 25: Subsequences; limit of a subsequence of a convergent sequence; any sequence has monotone subsequence; subsequential limits. Section 2.11.

Tuesday, October 26: lim sup, lim inf and subsequential limits. Section 2.11.

Wednesday, October 27: Theorem on lim sup of product. Ratios and n-th roots. Section 2.12. New homework!

Friday, October 29: Series; convergence and absolute convergence; Cauchy criterion; terms of convergent series tend to zero. Section 2.14.

Monday, November 1: Series: absolutely convergent series converge; comparison test; ratio test and root test. Section 2.14.

Tuesday, November 2: Examples of convergent and divergent series; integral test. Sections 2.14, 2.15.

Wednesday, November 3: More on integral test. Test for alternating series. Section 2.15. New homework! We will have QUIZ on Friday, November 5!

Friday, November 5: Two definition of continuity and their equivalence. Section 3.17. We had quiz today. Here are the solutions.

Monday, November 8: Quiz solutions; examples of continuous and discontinuous functions. Section 3.17. IMPORTANT: we will have another quiz on Wednesday, November 10 and the midterm on Friday, November 19.

Tuesday, November 9: Sum, product etc of continuous functions is continuous. Polynomial and rational functions are continuous. Superposition functions are continuous. Section 3.17.

Wednesday, November 10: Continuous functions are bounded and assume their maximum and minimum on closed intervals. Section 3.18. We had quiz today. Here are the solutions. New homework!

Friday, November 12: Intermediate value theorem and applications. Section 3.18.

Monday, November 15: Strictly increasing functions; continuity of inverse function. Section 3.18.

Tuesday, November 16: Uniform continuity. A continuous function on a closed interval is uniformly continuous. Section 3.19. REMINDER: We will have midterm on Friday, review is tomorrow!

Wednesday, November 17: Review before midterm.

Friday, November 19: We had midterm today. Here are the solutions. New homework!

Monday, November 22: Criteria for uniform continuity. Section 3.19.

Tuesday, November 23: Limits of functions. Section 3.20.

Wednesday, November 24: Power series, radius of convergence. Section 4.23.

Monday, November 29: Uniform convergence. Limit of uniformly convergent sequence of continuous functions is continuous. Section 4.24. We will have QUIZ on Friaday!

Tuesday, November 30: A power series represents continuous function in the interior of its convergence interval. Sections 4.25, 2.26.

Wednesday, December 1: Review before final.

Friday, December 3: Review before final. We had quiz today. Here are the solutions.

Practice exam is now available! Solutions are here.