Welcome to Math 409 – fun with epsilon!
In this fundamental course, we will revisit calculus, but this time with a mathematician’s hat, with rigorous proofs instead of hand-wavy arguments. This is pure mathematics at its finest, and should appeal to budding mathematicians like you. Enjoy!
Syllabus
Math 409 – Syllabus
Lecture Notes and Videos
| # |
Section |
Title |
Videos |
| 1 |
1, 2 |
Natural and Rational Numbers |
What is a number? |
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What is induction? |
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Induction Example |
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sqrt(2) is irrational |
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Algebraic Numbers |
| 2 |
2, 3 |
Rational and Real Numbers |
What is a field? |
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Ordered Field |
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Triangle Inequality |
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Rational Roots Theorem (optional) |
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Rational or Irrational (optional) |
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Rational Roots Theorem Proof (optional) |
| 3 |
4 |
Completeness Axiom (I) |
Max and Min |
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Upper Bound |
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What’s Sup? |
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Infimum |
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Least Upper Bound Property |
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inf(S) = -sup(-S) |
| 4 |
4, 5, 7 |
Completeness (II)Limits (I) |
Archimedean Property |
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Q is dense in R |
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What is Infinity? |
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Sequences |
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Limit Definition |
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Construction of R (optional) |
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Least Upper Bound Proof (optional) |
| 5 |
8 |
Limits (II) |
Example 1 |
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Example 2 |
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Example 3 |
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Example 4 |
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Example 5 |
| 6 |
9 |
Limit Theorems for Sequences |
Limits are Unique |
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Convergent Sequences are Bounded |
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Sum of Limits |
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Bounded Away from Zero |
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Product of Limits |
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Quotient of Limits (Ex 7) |
| 7 |
9, 10 |
Infinite Limits Monotone Sequences |
Example 8 |
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Example 9 |
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Example 10 |
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Infinite limit laws |
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Duality Theorem |
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Monotone Sequence Theorem |
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Decimal Expansions (optional) |
| 8 |
10 |
Limsup |
Limsup |
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Limsup vs Limit |
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Limsup Squeeze Theorem |
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Cauchy Sequences |
| 9 |
9,10 |
Cauchy Sequences
Subsequences (I) |
Completeness |
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Subsequence |
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Inductive Construction 1 |
| 10 |
1-10 |
Midterm 1 |
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| 11 |
11 |
Subsequences (II) |
Inductive Construction 2 |
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Monotone Subsequence |
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Bolzano-Weierstrass Theorem |
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Limit Points |
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Limsup and Subsequences (optional) |
| 12 |
12 |
Limsup Properties |
Limit Points are Closed |
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Limsup Product Rule |
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Pre-Ratio Test |
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13 |
Topology |
Topology Playlist |
| 13 |
14 |
Series (I) |
Partial Sums |
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Geometric Series |
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Cauchy Criterion |
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Comparison Test |
| 14 |
14, 15 |
Series (II) |
Root Test Proof |
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Ratio Test Proof |
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Ratio vs Root Test |
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Root Test Pitfall |
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Integral Test 1 |
| 15 |
15, 17 |
Series (III) Continuous Functions (I) |
Integral Test 2 |
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Alternating Series Test |
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Example 1 |
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Example 2 |
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Example 3 |
| 16 |
17 |
Continuous Functions (II) |
Equivalence |
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f+g is continuous |
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fg is continuous |
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f/g is continuous |
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g o f is continuous |
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Max is continuous |
| 17 |
18 |
Properties of Continuity |
Bounded Functions |
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Extreme Value Theorem Proof |
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Intermediate Value Theorem Proof |
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Fixed Point |
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What is a square root |
| 18 |
19 |
Uniform Continuity (I) |
Image of an Interval |
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Proof of Facts |
Continuity and Monotonicity |
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f inverse is continuous |
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Uniform Continuity |
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Example 1 |
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Example 2 |
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Continuity on [a,b] |
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Not Unif Cont (optional) |
| 19 |
19, 20 |
Uniform Continuity (II)Limits (I) |
Uniform Continuity and Derivatives |
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Uniform Continuity and Cauchy |
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Continuous Extensions |
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Example 1 |
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Example 2 |
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Example 3 |
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Example 4 |
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Example 5 |
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Example 6 |
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Example 7 |
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21, 22 |
More Topology |
Topology Playlist |
| 20 |
11-19 |
Midterm 2 |
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| 21 |
20, 28 |
Limits (II)Derivatives |
Product Rule Proof |
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Quotient Rule |
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Chain Rule Proof |
| 22 |
29 |
Mean Value Theorem |
Rolle’s Theorem Proof |
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Mean Value Theorem Proof |
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MVT and Fixed Points |
| 23 |
30 |
L’Hopital’s Rule |
L’Hopital’s Rule Proof |
| 24 |
32 |
Riemann Integral (I) |
Darboux Integral |
| 25 |
32, 33 |
Riemann Integral (II)Integral Properties (I) |
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| 26 |
33 |
Integral Properties (II) |
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| 27 |
34 |
The Fundamental Theorem of Calculus |
FTC 2 Proof |
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FTC 1 Proof |
| 28 |
|
Final Exam Review |
Essence of Analysis |
|
1-34 |
Final Exam, W Dec 15, 8-10 am |
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Homework
Here is the schedule of homework, which is always due on Fridays at 11:59 pm (Texas Time). You will submit the homework on Canvas
| Homework |
Date |
Solutions |
Comments |
| Homework 1 |
09/03 |
AP Solutions |
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|
Book Solutions |
1.9b |
| Homework 2 |
09/10 |
AP Solutions |
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Book Solutions |
3.8, 4.14a |
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Reverse Triangle Inequality |
3.5 video |
|
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sup(A+B) = sup(A) + sup(B) |
4.14 video |
|
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What is Q? |
AP1 video |
|
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Algebraic Numbers are Countable |
AP5 video |
| Homework 3 |
09/17 |
AP Solutions |
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Book Solutions |
4.16, 7.4, 8.4, 8.5b, 8.7b, 8.8a, 8.9a, 8.10 |
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Squeeze Theorem Proof |
8.5a video |
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Example 6 |
AP video |
| Homework 4 |
09/24 |
AP Solutions |
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Book Solutions |
9.12a, 10.8 |
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Golden Limit |
AP3 video |
|
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Babylonian Square Root |
AP6 video |
| Homework 5 |
10/08 |
AP Solutions |
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Book Solutions |
11.8, 11.9, 11.11, 12.4 |
|
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Integers are Complete |
AP1 video |
|
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Cauchy Construction of R |
AP2 video |
|
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Sequence with Further Subsequence |
AP3 + AP4 video |
|
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Limsup of Sum |
12.4 video |
| Homework 6 |
10/15 |
AP Solutions |
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Book Solutions |
12.8, 12.12, 12.14, 14.6(a) |
|
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Averages |
12.12 video |
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Limit with n! |
12.14 video |
|
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Sum n/2^n |
AP4(c) video |
|
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Diagonally telescoping sum |
AP4(d) video |
|
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Block Test 1 |
AP5(a)(b) video |
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Block Test 2 |
AP5(c) video |
|
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Block Test 3 |
AP5(d) video |
|
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(1+1/n)^n the COOL way |
AP6 video |
| Homework 7 |
10/22 |
AP Solutions |
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Book Solutions |
17.9 + 17.10 |
|
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(a_n)^2 converges |
15.6 video |
|
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Example 4 |
17.9(d) Video |
|
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Cauchy-Schwarz |
AP2(a) video |
|
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The Euler-Mascheroni Constant |
AP3 video |
|
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Limit Comparison Test |
AP4 video |
|
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e is irrational |
AP5 video |
|
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Sum 1/n^2 |
AP6 video |
| Homework 8 |
10/29 |
AP Solutions |
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Book Solutions |
17.8(b) + 17.12 + 17.13 + 17.14 + 18.9 + 18.10 + 18.12 |
|
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Popcorn Function |
17.14 Video |
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Example 5 |
AP1(a) Video |
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Example 6 |
AP1(b) Video |
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Example 7 |
AP1(c) Video |
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Continuity without epsilon |
AP2 + AP3 Video |
|
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Press fff to pay respects |
AP4 Video |
| Homework 9 |
11/12 |
AP Solutions |
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Book Solutions |
19.2 + 19.4 + 19.5 + 19.6 + 19.8 + 20.11 + 20.20 |
|
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Squeeze Theorem |
AP2 Video |
|
|
sin(x)/x |
AP3 Video |
| Homework 10 |
11/19 |
AP Solutions |
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|
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Book Solutions |
28.6 + 28.15 + 29.4 + 29.5 + 29.18 + 30.6 + 30.7 |
|
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Banach Fixed Point Theorem |
29.18 Video |
|
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Hopital Exercise |
30.6 Video |
|
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Hopital Counterexample |
30.7 Video |
|
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Cool Quotient Rule |
AP2 Video |
|
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Exponential Properties |
AP4 Video |
|
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Don’t use L’Hopital |
AP5 Video |
| Homework 11 |
12/03 |
AP Solutions |
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Book Solutions |
32.1 + 33.4 + 33.7 + 33.8 + 33.10 + 34.8 + 34.10 |
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Integral f inverse |
34.10 Video |
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Integration Sucks |
AP 1 Video |
|
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Stieltjes Integral |
AP 2 Video |
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Product Integral |
AP 3 Video |
YouTube Playlists and more
Here are the YouTube playlists for the first three chapters of the course.
Exams
Here you can find info about exams, as well as other goodies such as practice exams.
Midterm 1: Thursday, September 30, covers sections 1-10 (except 6 and Cauchy Sequences)
Midterm 2: Thursday, November 4, covers sections 10-19 (except 13, 16)
Final Exam: Wednesday, December 15, 8-10 am, covers the entire course