• Preliminaries

• Different Number Systems and Some Fine Properties of the Reals (including a geometric proof of the irrationality of the square root of 2)
• Proof that both the rationals and the irrationals are dense in the reals
  
• Absolute Value and Inequalities
• The Binomial Theorem
• Reviewing Proof by Induction with three examples

Homework 1    (Due Monday the 20th of August)

Homework 1 with Solutions

• Sequences

• Definition, Notations, and Examples

• Boundedness and Monotonicity
  
• Boundedness
• Monotonicity (Increasing and Decreasing)
  
• Two Examples of proving a sequence is bounded and monotone

• Convergence
• Definition and five worked examples
• (-1)^n is divergent (Did not completely discuss in class - see lecture on subsequences below for an alternative proof)

Sample Practice Questions 2  (with solutions)

Homework 2    (Due Wednesday the 29th of August)
 

• Properties of Convergent Sequences
  
• Limits are unique
• Convergence implies Bounded
• Order Limit Law

• Tools for Computing Limits
• The "Baby Squeeze Theorem" and applications
• The "Ratio Test for Sequences" and applications    [* We didn't discuss this - so non-examinable]
• Limit Laws (including proofs and some applications, including the "Full Squeeze Theorem")
  
• Special Limits (including further applications of limit laws)
  

Sample Practice Questions 3  (with solutions)

Homework 3    (Due Wednesday the 5th of September)

• Subsequences
• Definition, inherited properties, and the "Rising Sun Lemma"  [* Proof of "Rising Sun Lemma" is non-examinable]

• What is Reality?
• Least Upper Bounds, Greatest Lower Bounds, and the Axiom of Completeness (AoC)
• Example of a non-empty set of rationals that is bounded above, but has no rational least upper bound, and
  a proof of the existence of any root of any positive real in the reals      [* Statements only for now, we will revisit this example]
  
• The Monotone Convergence (MCT) and Bolzano-Weierstrass (BW) Theorems
  
• Some examples of using the MCT
• Limit Superior and Limit Inferior (see Sample Practice Questions 4 - Question 5, below)
• Discussion of the "Nested Interval Property" and equivalence of AoC, MCT, and BW    [* We didn't discuss this - so non-examinable]
• Cauchy Sequences and the Cauchy Criterion (CC)
  

Sample Practice Questions 4  (with solutions)

Homework 4    (Due Monday the 17th of September)

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• Exam 1 Materials
  

• Exam 1 Study Guide  
  

• Sample Old Exams:  
• Sample Exam 1 - Version 1   (Version 1 with Solutions)
  
• Sample Exam 1 - Version 2   (Version 2 with Solutions)
• Sample Exam 1 - Version 3   

• Exam 1
  

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• Infinite Series

• Summary (Overview of the results we will establish) 

• Definition, Notation, and Examples
• A "Test for Divergence", the Cauchy Criterion and a second proof, and further examples

• Series on non-negative terms
• Monotone Convergence Theorem on Series
  
• Direct and Limit Comparison Tests
  
• Cauchy Condensation and variations on p-series [* Proof of "Cauchy Condensation" is non-examinable]
• Ratio Test (Calculus version)
  

Homework 5    (Due Monday the 1st of October)

Homework 5 with Solutions

• Series with both positive and negative terms
• Absolute convergence implies convergence (two proofs)
  
• Alternating Series Test and the notion of Absolute and Conditional Convergence
• The Ratio and Root Tests

Homework 6    (Due Friday the 5th of October)

Homework 6 with Solutions

• The Integral Test, Euler's constant, and the sum of the alternating harmonic series   [* Proofs are non-examinable]

• Rearrangements   [* Proofs are non-examinable]

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• Power Series and Continuity

• Power Series
• Radius and Interval of Convergence
• Examples

• Definition of Continuity

• Sequential Characterization
• Operations with continuous functions
• Examples of discontinuous functions
  

Homework 7    (Due Friday the 12th of October)

Homework 7 with Solutions

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• Exam 2 Materials
  

• Exam 2 Study Guide  
  

• Sample Old Exams:  
• Sample Exam 2 - Version 1   (Version 1 with Solutions)
  
• Sample Exam 2 - Version 2   (Version 2 with Solutions)
  

• Exam 2

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• Continuity and Differentiation
  

• Continuity and the Intermediate and Extreme Value Theorems
  

• Functional Limits (including the Sequential Characterization)

• Differentiation
• Interior Extrema Theorem
  
• Rolle's Theorem and the (Generalized) Mean Value Theorem
• L'Hopital's Rule (only proved the "0/0" case)
  

Homework 8    (Due Friday the 2nd of November)

Homework 8 with Solutions

• Power Series and Taylor Series

• Power Series can be differentiated (and integrated) term-by-term inside their radius of convergence 
       [* Only the proof that the original and differentiated series have the same radius of convergence is examinable]

• Lagrangian Remainder Estimate for Maclaurin Series and Applications

• Alternating Series Remainder Estimate

• Examples
  

Homework 9    (Due Friday the 9th of November)

Homework 9 with Solutions

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• Exam 3 Materials
  

• Exam 3 Study Guide  
  

• Sample Old Exams:  
• Sample Exam 3 - Version 1   (Version 1 with Solutions)
  
• Sample Exam 3 - Version 2   (Version 2 with Solutions)

• Exam 3

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• Uniform Convergence of Sequences and Series of Function
  

• Pointwise and Uniform Convergence of Sequences of Functions
• Examples

• Uniform Convergence preserves Continuity
  

• Uniform Convergence of Series of Functions
• Cauchy Criterion for Uniform Convergence [and test for non-uniform convergence of series]
  
• Weierstrass M-Test
• Examples
  

Homework 10    (Due Monday the 3rd of December)

Homework 10 with Solutions

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• Final Exam Materials

•  Practice Final Exam
  

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