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Math 315 FALL 2004, List of lectures
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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.