Elementary Real Analysis
Fall `07 - Humke`s Version

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Below is a tentative list of Assignments, ERA Words, ERA Facts and More

Chapter 1

Chapter 1 Chapter 1 Chapter 1 Chapter 1

Chapter 2

Chapter 2

  • Section 2.3 ( Note the order change)
    1. Definitions
      • subsequence
      • limit inferior
      • limit superior
      • subsequential limit
    2. Theorems
      • {x_n} converges if and only if every subsequence converges
      • Every sequence has a monotone subsequence
      • Bolzano-Weierstrass Theorem
    3. Gordon Problems: 1-4, 6,9,13,14,17,26,27 (Assigned ; Due )

Chapter 2
  • Section 2.2
    1. Definitions
      • Cauchy sequence
      • A nest of inervals (or a nested sequence of intervals)
      • A potpouri of convergent sequences theorem
    2. Theorems
      • If {a_n} is monotone, then {a_n} convergese <=> {a_n} is bounded.
      • Cauchy sequences are bounded
      • A sequence, {a_n} converges <=> {a_n} is Cauchy
      • Nested Interval Theorem
    3. Gordon Problems: 1,3,4,5,9,10,14,16,18,24

Chapter 3
  • Section 3.1
    1. Definitions
      • limit of a function at a point, x_o (2 definitions)
      • right and left hand limits of a function at a point, x_o
    2. Theorems
      • Algebra of Limits Theorem for Functions (SAL)
      • Squeeze Theorem for Functions
    3. Gordon Problems 2,6,9,10,13, 23,24,27,28,32 (Assigned ; Due )

Chapter 3
  • Section 3.2
    1. Definitions
      • f is continuous at x_o (2 definitions)
      • f:A->B is continuous
      • f has a jump discontinuity at x_o
      • f has a removable discontinuity at x_o
    2. Theorems
      • Algebra of Continuous Functions Theorem (CAL)
      • Polynomials are everywhere continuous
      • Rational Functions are continuous except at poles.
    3. Gordon Problems 1,4,6,7,8,9,10,13,16,18 (Assigned ; Due )

Chapter 3
  • Section 3.3
    1. Definitions
      • intermediate value of a function f:A->B
      • f has the intermediate value property on a set A
      • extreme value of a function f:A->B
      • f is locally bounded
      • the set E is compact
    2. Theorems
      • Intermediate Value Theorem
      • Extreme Value Theorem
      • If the domain of a continuous function is compact, so is the range
      • If the domain of a continuous function is an interval, so is the range
      • If f is monotone, f has only jump discontinuities
      • If f is monotone, f has at most countably many discontinuities
      • A monotone function with the intermediate value property is continuous.
      • A continuous strictly monotone function has a continuous strictly monotone inverse.
    3. Gordon Problems 1,3,5,6,7,11,13,16,18,27,34,39 (Assigned ; Due )

Chapter 4
  • Section 4.1
    1. Definitions
      • difference quotient
      • f is differentiable at x_o
      • f is differentiable on and interval [a,b]
    2. Theorems
      • Algebra of Differentiable Functions Theorem
        1. constant multiplier rule
        2. sum (or difference) rule
        3. product rule
        4. quotient rule
        5. humke rules
        6. chain rule
      • Derivatives of Standard Functions Theorem
    3. Gordon Problems 1,2,4,6,10 (use MI), (Assigned ; Due ) 12,16,2122,23,24 (Assigned ; Due )

Chapter 4
  • Section 4.2
    1. Definitions
      • critical point
    2. Theorems
      • Rolle's Theorem
      • Mean Value Theorem
      • Monotonicity Theorem for Derivatives
      • First Derivative Test
      • Second Derivative Test
      • L'Hosptal's Rule
    3. Gordon Problems 2,3,4,8,10,17,20,23 (Assigned ; Due )

Chapter 7
  • Section 7.1
    1. Definitions
      • {fn}->f pointwise on A
      • {Sigma fn}->f pointwise on A
    2. Gordon Problems 7.1A page 243 Examples 1-8 investigate and verify (Assigned ; Due )
    3. Gordon Problems 7.1B page 246 Problems 1-5,7,8,15 (Assigned ; Due )

Chapter 7
  • Section 7.2
    1. Definitions
      • {fn}->f uniformly on A
      • {Sigma fn}->f uniformly on A
    2. Theorems
      • If {fn}->f pointwise on an interval I and Mn=sup{|fn(x)- f(x)| : x in I}. Then {fn}->f uniformly on I if and only if {Mn}->0.
      • Dini's Theorem
    3. Gordon Problems 7.2 page 251 Problems 1,3,4,6,7,10 (Assigned ; Due )

Chapter 5
  • Section 5.1
    1. Definitions
      • partition, norm of a partition, refinement, tagged partition, Riemann integrable function, Riemann integral, oscillation of a function on an interval
    2. Theorems
      • Algebra of Riemann integrable functions theorem
    3. Gordon Problems Page 169 1,3,5,6,10, 16,17 (Assigned ; Due )

  • Section
    1. Definitions
    2. Theorems
    3. Gordon Problems


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