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Math 315 WINTER 2008, List of lectures
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Monday, January 7: Peano axioms for natural numbers; principle
of mathematical induction. Section 1.1.
Tuesday, January 8: Integer and rational numbers. Square root of 2 is
irrational. Algebraic numbers and Theorem on rational solutions of
algebraic equations. Section 1.2.
Wednesday, January 9: Axioms of ordered field and consequences.
Section 1.3.
Friday, January 10: Absolute values. Completeness axiom. Sections 1.3, 1.4.
Monday, January 14: Completeness axiom fails for rational numbers.
Section 1.4.
Tuesday, January 15: Consequences of the completeness axiom. Infinite
supremum and infinum. Real numbers from rationals. Sections 1.4, 1.5, 1.6.
We will have QUIZ next Tuesday!!!
Wednesday, January 16: Sequences, definition of limit, limit is unique
if exists. Section 2.7. New homework!
Friday, January 18: Examples of formal proofs for limit calculations.
Section 2.8.
Tuesday, January 22: Limit of square root sequence; any convergent
sequence is bounded; multiplication by a scalar. Sections 2.8, 2.9.
We had quiz today.
Here are the solutions.
Wednesday, January 23: Limit of the sum of convergent sequences; limit of
the product and the ratio of convergent sequences. Section 2.9.
New homework!
Friday, January 25: Standard examples of limits. Infinite limits.
Section 2.9.
Monday, January 28: Monotone sequences have limits. Section 2.10.
We will have MIDTERM on next Monday.
Tuesday, January 29: Upper limit (lim sup)
and lower limit (lim inf). Section 2.10.
Wednesday, January 30: lim sup=lim inf is the same as existense of lim.
Cauchy sequences. Section 2.10. New homework!
Friday, February 1: Convergent sequences = Cauchy sequences. Section 2.10.
Monday, February 4: We had midterm today.
Here are the solutions.
Tuesday, February 5: Subsequences; limit of a subsequence of a convergent
sequence; any sequence has monotone subsequence; subsequential limits.
Section 2.11.
Wednesday, February 6: lim sup, lim inf and subsequential limits.
Section 2.11.
Friday, February 8: Theorem on lim sup of product. Ratios and
n-th roots. Section 2.12.
New homework!
Monday, February 11: Series; convergence and absolute convergence;
Cauchy criterion; terms of convergent series tend to zero.
Section 2.14. We will have QUIZ on Friday!!!
Tuesday, February 12: Series: comparison test; ratio test and root test.
Section 2.14.
Wednesday, February 13: Examples of convergent and divergent series;
integral test. Sections 2.14. New homework!
Friday, February 15: Integral test and test for alternating series.
Section 2.15. We had quiz today.
Here are the solutions.
Monday, February 18: Two definition of continuity and their equivalence.
Section 3.17.
Tuesday, February 19: Examples of continuous and
discontinuous functions. Section 3.17.
Wednesday, February 20: Sum, product etc of continuous functions is
continuous. Polynomial and rational functions are continuous. Superposition
functions are continuous. Section 3.17. New homework!
Friday, February 22: Continuous functions are bounded and assume
their maximum and minimum on closed intervals. Section 3.18.
Monday, February 25: Intermediate Value Theorem and its consequences.
Section 3.18. We will have MIDTERM next Monday!!!
Tuesday, February 26: More consequences for Intermediate Value Theorem.
Section 3.18.
Wednesday, February 27: Review for midterm.
New homework!
Friday, February 29: Inverse functions and their continuity. Section 3.18.
Monday, March 3: We had midterm today.
Here are the solutions.
Tuesday, March 4: Uniform continuity. A continuous function on
a closed interval is uniformly continuous. Limits of functions. Sections 3.19-3.20.
Wednesday, March 5: Power series, radius of convergence. Section 4.23.
Review problems are here!
Friday, March 7: Uniform convergence. Sections 4.24-4.25.
Monday, March 10: There will be no lecture.
Tuesday, March 11: A power series represents continuous function
in the interior of its convergence interval. Section 4.26.
Wednesday, March 12: There will be no lecture.
Friday, March 14: Review.
Solutions for review
problems are here.