Mathematics 310 Homepage

310 INTRODUCTION TO ANALYSIS I (3+0) 3 credits

An examination of the theory of calculus of functions of one variable with emphasis on rigorously proving theorems about real numbers, convergence, continuity, differentiation and integration. Prereq(s): MATH 283.

Instructor  Course Section                       Time
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Eric Olson  001 Math 310 INTRO TO ANALYSIS I     MWF 9:00-9:50am PE103

Course Information

Instructor:

Eric Olson

email:

Please contact me through WebCampus

Office:

DMS 238 MW from 11am to 1pm

Homepage:

https://fractal.math.unr.edu/~ejolson/310/

Live Stream:

If you can't come to class due to sickness, quarantine or other reasons, please join via the Zoom link in WebCampus.

Primary Text:

Frank Dangello, Michael Syfried, Introductory Real Analysis, Houghton Mifflin Company, 2000.

Other Reading:

Joseph Taylor, Foundations of Analysis, Twelfth Edition, American Mathematical Society, 2012.

Jay Cummings, Real Analysis, A Long-Form Mathematics Textbook, 2019.

Gerald Folland, Advanced Calculus, First Edition, Pearson.

Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976.

Robert C. Wrede, Murray Spiegel, Schaum's Outline of Advanced Calculus, Second Edition, McGraw-Hill.

Student Learning Outcomes

• Describe the real line as a complete, ordered field.
• Determine the basic topological properties of subsets of the real numbers.
• Use the definitions of convergence to approximate by sequences, series, and functions.
• Determine the continuity, differentiability, and integrability of functions defined on subsets of the real line.
• Apply the Mean Value Theorem and the Fundamental Theorem of Calculus to problems in the context of real analysis.
• Understand definitions and produce rigorous proofs of results that arise in the context of real analysis.
• Write solutions to problems and proofs of theorems that meet rigorous standards based on content, organization and coherence, argument and support, and style and mechanics.

Class Handouts

Homework

HW1 due Friday, Feb 2 (solutions)

Turn in 1.1#1

Practice 1.1#7, 1.2#4, 1.2#10, 1.2#11, 1.3#1, 1.3#2

HW2 due Friday, Feb 9 (solutions)

Turn in 1.4#10

Practice 1.3#8, 1.3#9, 1.4#2, 1.4#5, 2.1#2, 2.1#6

HW3 due Friday, Feb 16 (solutions)

Turn in 2.2#3

Practice 2.1#7, 2.2#1abcd, 2.2#7, 2.2#8, 2.2#9

HW4 due Friday, Feb 23 (solutions)

Turn in 2.3#6

Practice 2.3#3, 2.3#4, 2.4#2, 2.4#4, 2.4#10, 2.4#15

HW5 due Friday, Mar 1 (solutions)

Turn in 3.1#5

Practice 3.1#2, 3.1#3, 3.1#9cd, 3.2#1ef, 3.2#2ab, 3.2#6

HW6 due Friday, Mar 15 (solutions)

Quiz 3 on Monday Mar 18 over homework

Practice 3.3#1abc, 3.3#3, 3.3#4, 3.4#2, 3.4#3, 3.5#2, 3.5#3

HW7 due Friday, Apr 5 (solutions)

Quiz 4 on Monday Apr 8 over homework

Practice 3.6#2, 3.6#6, 3.6#7, 3.7#1d, 3.7#6, 3.8#1c, 3.8#4

HW8 due Wednesday, May 1 (solutions)

Quiz 5 on Friday May 3 over homework

Practice 4.3#6, 4.4#6, 4.5#5, 4.6#10, 5.2#1abc, 5.2#3, 5.3#7

Lecture Notes

• Lecture 1: Mathematics as a Language
• Lecture 2: Indirect Proofs
• Lecture 3: Sets and DeMorgan's Law

• Lecture 4: Functions
• Lecture 5: Induction
• Lecture 6: The Real Numbers

• Snow day: no notes
• Lecture 7: The Completeness Axiom
• Lecture 8: Density of the Rationals

• Lecture 9: Cardinality
• Lecture 10: Countably Infinite
• Lecture 11: Countable Unions

• President's Day: no notes
• Lecture 12: Uncountable Reals
• Lecture 13: Convergence

• Lecture 14: Limit Laws
• Lecture 15: Subsequences
• Lecture 16: Monotone Convergence

• Delayed Start: Exam moved to Wednesday
• Exam 1: no notes
• Lecture 17: Bolzano-Weierstrass Theorem

• Lecture 18: Bolzano-Weierstrass for sets
• Lecture 19: Cauchy Sequences
• Lecture 20: Limit Superior and Inferior

• Lecture 21: Exercise 3.4#9
• Lecture 22: Continuity
• Lecture 23: Continuity...

• Lecture 24: Continuity and Sequences
• Lecture 25: Continuity and Sequences...
• Lecture 26: Consequences of Continuity

• Lecture 27: Consequences of Continuity...
• Lecture 28: Uniform Continuity
• Lecture 29: Uniform Continuity...

• Exam 2: no notes
• Lecture 30: Monotone Functions
• Lecture 31: Derivatives

• Lecture 32: f'(x) has IVP
• Lecture 33: Mean Value Theorem
• Lecture 34: Riemann Integration

• Lecture 35: Riemann Sums
• Lecture 36: More Riemann Sums
• Lecture 37: Properties of Integral

• Lecture 38: Continuous Functions are Integrable

Announcements

[15-May-2024] Final Exam

Please review the following in preparation:

1. All definitions from
   • Chapters 1 – 4
   • Sections 5.1 – 5.3
   • Sections 6.1 – 6.4
2. The homework problems.
   • Section 1.3#1 Prove f(A∪B) = f(A)∪f(B).
   • Section 1.3#9 Find a bijection from (0,∞) onto (0,1).
   • Section 1.4#2 Show 1³+2³+ . . . + n³=(n(n+1)/2)².
   • Section 1.4#5 Prove n divides 7ⁿ − 3ⁿ.
   • Section 1.4#10 Bernoulli's Inequality.
   • Section 2.1#2 Find x and y irrational such that xy∈Q.
   • Section 2.1#6 Write 0.33474747... as a fraction.
   • Section 2.2#1abcd Find the supremum and infimum of the following sets.
   • Section 2.2#9 In general show sup(A) sup(B) ≠ sup {ab : a∈A and b∈B}
   • Section 2.3#3abcd Find the supremum and infimum of the following sets.
   • Section 2.3#4 The irrational numers are dense in R.
   • Section 2.3#6 sup{f(x)+g(x) : x∈X} ≤ sup{f(x) : x∈X} + sup{g(x) : x∈X}.
   • Section 2.4#2 Let f:A→B be one-to-one. If B is finite, show A is finite.
   • Section 2.4#10 Let A be uncountable and B⊆A be countable. Show A\B is uncountable.
   • Section 3.1#5 If x_n≥0 and x_n→x, show that √x_n→√x.
   • Section 3.2#1ef Find the limits of the following sequences.
   • Section 3.2#2ab Find examples of two sequences.
   • Section 3.2#6 If x_n is bounded and y_n→0, show x_ny_n→0.
   • Section 3.3#1abc Can you give an example of a sequence?
   • Section 3.4#2 Let x₁=3 and x_n+1=2−(1/x_n).
   • Section 3.4#3 Let x₁=√2 and x_n+1=√(2+x_n).
   • Section 3.5#2 Let x_n be a bounded sequence of integers. Show x_n has an eventually constant subsequence.
   • Section 3.5#3 Show a nonconvergence bounded sequence has two subsequences each with a different limit.
   • Section 3.6#2 Let x_n=1+1/2+1/3+ . . . + 1/n. Show x_n is not Cauchy.
   • Section 3.7#1d Establish the limit n-6√n → ∞.
   • Section 3.7#6 Suppose x_n/y_n→L where 0<L<∞. Show x_n→∞ if and only if y_n→∞.
   • Section 3.8#1c Find limsup x_n and liminf x_n.
   • Section 3.8#4 Show liminf a_n + liminf b_n ≤ liminf (a_n+b_n).
   • Section 4.3#6 Let f:D→R such that f(x)→L as x→c. Find a neighborhood U of c on which f is bounded.
   • Section 4.5#5 Show f(x)=1/x² is uniformly continuous on [1,∞) but not on (0,1].
   • Section 4.6#10 Let f be a monotone function satisfying the intermediate value property. Show f is continuous.
   • Section 5.2#1abc Application of mean value and intermedate value theorems.
   • Section 5.2#3 Suppose f is differentiable and has n distinct roots. Show f' has at least n−1 distinct roots.
   • Section 5.3#7 If x>0 show that 1+x+x²/2 < e^x < 1+x+x²e^x/2.
3. Know the proofs of
   • Theorem 1.1 √2 is irrational.
   • Proposition 2.2 The Triangle Inequality.
   • Theorem 2.2 Q is dense in R.
   • Theorem 2.5 R is uncountable.
   • Proposition 3.2 A convergent sequence is bounded.
   • Theorem 3.2 Limit Laws for Sequences.
   • Theorem 3.7 Monotone Convergence Theorem.
   • Theorem 3.10 Bolzano-Weierstrass Theorem for Sequences.
   • Proposition 3.5 A convergent sequence is Cauchy.
   • Theorem 3.12 If a sequence is Cauchy then it converges.
   • Theorem 4.2 A continuous function on [a,b] has an absolute maximum.
   • Proposition 4.3 The composition of continuous functions is continuous.
   • Proposition 4.9 The discontinuities of a monotone function are countable.
   • Theorem 4.4 A continuous function on [a,b] is uniformly continuous.
   • Theorem 4.5 Uniformly continuous functions map Cauchy sequences to Cauchy sequences.
   • Theorem 5.1 If f is differentiable at c then it's continuous at c.
   • Proposition 5.1 The rules of differential calculus parts 1 and 2.
   • Theorem 5.2 The Chain Rule.
   • Proposition 5.2 An interior point c which is a local extrema has f'(c)=0.
   • Theorem 5.4 Generalized Mean Value Theorem
   • Theorem 6.2 f∈R[a,b] if and only if ∀ε>0 ∃P∈P[a,b] such that U(P,f)−L(P,f)<ε.
   • Proposition 6.3 Linearity of the integral.
   • Proposition 6.4 Monotonicity of the integral.
   • Theorem 6.7 Continuous functions are in R[a,b].
   • Theorem 6.9 Monotone functions are in R[a,b].
4. Be able to state (without proof) exactly
   • Theorem 3.9 Monotone Subsequence Theorem.
   • Theorem 3.10 Bolzano-Weierstrass Theorem for Sequences.
   • Theorem 4.3 Intermediate Value Theorem.
   • Theorem 5.6 Taylor's Theorem.
5. Examples
   • A function that is continuous on (a,b) but not uniformly continuous.
   • A bounded function f:[a,b] → R such that f ∉ R[a,b].
   • A function f:X → Y and a set A ⊆ X such that f ⁻¹(f(A)) ≠ A.
   • A function f:X → Y and a set B ⊆ Y such that f(f ⁻¹(B)) ≠ B.
   • A bounded sequence x_n and a convergent sequence y_n such that x_ny_n does not converge.
   • A continuous function h:R → R such that h((-1,1)) is not an open interval.

[03-May-2024] Quiz 5

[15-Apr-2024] Exam 2

• The first seven homework assignments.
• Definitions in the book up to Chapter 4.5
  • Make sure you know the definitions of convergence, continuity and uniform continuity.
• Proofs from the book up to Chapter 4.1
  • Exercise 4.1#4: Let f:D→R be continuous at c in D. Let h be in R with f(c)<h. Show that there is a neighborhood U of c such that if x is in U∩D, then f(x)<h.
  • Proposition 4.8: If f:[a,b]→R is continuous on [a,b], then f is bounded on [a,b].
• A surprise question related to the material.

[06-Apr-2024] Homework 7 Solutions

[16-Mar-2024] Homework 6 Solutions

[06-Mar-2024] Exam 1 (rescheduled)

• The first four homework assignments.
• Definitions in the book up to Chapter 3.2.
• Proofs from the book up to Chapter 3.2.
• A surprise question related to the material.

[04-Mar-2024] Exam 1 (cancelled due to snow)

• The first four homework assignments.
• Definitions in the book up to Chapter 3.2.
• Proofs from the book up to Chapter 3.2.
• A surprise question related to the material.

[02-Mar-2024] Homework 5 Solutions

[01-Mar-2024] Quiz 2

[20-Feb-2024] Homework 4 Solutions

[20-Feb-2024] Homework 3 Solutions

[16-Feb-2024] Quiz 1

[12-Feb-2024] Homework 2 Solutions

[08-Feb-2024] Homework 1 Solutions

[19-Jan-2024] Welcome Spring 2024

Mathematics 310 is the first course in the UNR mathematics curriculum where the emphasis is on mathematical proof and reasoning. This course focuses on a rigorous justification of the topics covered in Mathematics 181-283 and provides a stepping stone to higher-level mathematics. There will be homework assignments and quizzes weekly. Mathematical proofs should be carefully written using complete English sentences, proper grammar, spelling and punctuation. This is a hard course.

Do not come to class if you are sick--even if it's something other than COVID-19. If you are subject to quarantine because of exposure to a person who is sick, please stay home.

This section of Math 310 is in person. However, I will live-stream our class meetings each day at a link available in WebCampus for those who are sick or unable to attend on a particular day. I will also maintain an online archive of course materials including lecture notes, assignments and other announcements.

Tentative Course Schedule

#   Date     Chapter     Topic
------------------------------------------------------------------------
1   Jan 22    1.1        Proofs
2   Jan 24    1.2        Sets
3   Jan 26    1.3        Functions

4   Jan 29    1.4        Mathematical Induction
5   Jan 31    2.1        Algebraic and Order Properties of R
6   Feb 02    2.1        Properties of R continued...

7   Feb 05    ***        Snow Day
8   Feb 07    2.2        The Completeness Axiom
9   Feb 09    2.3        The Rational Numbers are Dense in R

10  Feb 12    2.4        Cardinality
11  Feb 14    2.4        Cardinality continued...
12  Feb 16               Quiz 1

    Feb 19    ***        President's Day
13  Feb 21    3.1        Convergence
14  Feb 23    3.2        Limit Theorems

15  Feb 26    3.3        Subsequences
16  Feb 28    3.4        Monotone Sequences
17  Mar 01    3.4        Monotone Sequences continued...

18  Mar 04               Exam I
19  Mar 06    3.5        Bolzano-Weierstrass Theorems
20  Mar 08    3.5        Bolzano-Weierstrass continued...

21  Mar 11    3.6        Cauchy Sequences
22  Mar 13    3.7        Limits at Infinity
23  Mar 15    3.8        Limit Superior and Limit Inferior

24  Mar 18    4.1        Continuous Functions
25  Mar 20    4.2        Limit Theorems
26  Mar 22    4.3        Limits of Functions

*** Spring Break Saturday Mar 23 to Sunday March 31

27  Apr 01    4.4        Consequences of Continuity
28  Apr 03    4.4        Consequences of Continuity continued... 
29  Apr 04    4.5        Uniform Continuity

30  Apr 08    4.6        Discontinuous and Monotone Functions
31  Apr 10    5.1        The Derivative
32  Apr 12    5.2        Mean Value Theorems

33  Apr 15               Exam II
34  Apr 17    5.3        Taylor's Theorem
35  Apr 19    5.3        Taylor's Theorem continued...

36  Apr 22    5.4        L'Hopital's Rule (skipped)
37  Apr 24    6.1        Existence of Riemann Integral
38  Apr 26    6.2        Riemann Sums

39  Apr 29    6.3        Properties of the Riemann Integral
40  May 01    6.3        Riemann Integral continued...
41  May 03    6.4        Families of Riemann Integrable Functions

42  May 06    ***        Review

*** Prep Day May 08
*** Final exam Wednesday, May 15 from 8:00-10:00am in PE103

Grading

     2 Exams                   50 points each
     5 Quizzes                  5 points each
     5 Homework Assignments     5 points each
     Final Exam               100 points
    ------------------------------------------
                              250 points total

    Grade        Minimum Percentage
      A                 90 %
      B                 80 %
      C                 70 %
      D                 60 %

Quiz and Exam Schedule

Course Policies

Communications Policy

Late Policy

Plagiarism

Diversity

Statement on Academic Success Services

Equal Opportunity Statement

Statement on Audio and Video Recording

Academic Conduct

Final Exam