Introduction to a College Course in Real Analysis

Purpose and Target Audience:

The project is primarily aimed at college students in the second or third year of a mathematics major.
The focus is a course in Analysis of Functions of a Real Variable (Real Analysis, for short. Sometimes also called Advanced Calculus).

Taking this course is a significant step on the path of mathematics, much like the step from arithmetic to Algebra or the step from Algebra to Calculus.
For students to even get to this class, they are likely university math majors, which suggests that they like math and have been good at it.
So I was suprised that when I took this course, 40% of the class got F's and 20% got D's.

So, what makes this course difficult?

• The objective of the course is not always clear.
  Unlike previous courses, the objective is not to introduce techniques for solving new types of equations.
  


• The organization of the course is not explained.
  So it can seem like a bunch of random topics with known (obvious) results.


• Study techniques which worked for previous courses, like Calculus, may not work on this course.


• Traditional styles of teaching this course (or, at least, the way I was taught) teach proofs primarily by showing examples of proofs.
  But often, the way a proof is written in the book doesn't show how the author came up with the proof.

I will give an overview of the course, followed by some examples of how to approach writing proofs.
I will end with a proposal for a new mindset needed to study this type of math.

Context for a Real Analysis Course

There are several early milestones along the path of mathematical education.

In elementray school through high school:

• Arithmetic introduced manipulating numbers.
• Algebra introduced abstraction -- manipulating variables (letters) in place of numbers.
• Calculus introduced manipulating functions.

In the first two years of college:

• Linear algebra and vector calculus moved the manipulations into multiple dimensions.
• Differential equations combined aspects of algebra and Calculus.
  Instead of finding numbers, x, that satisfy an equation involving x, the student finds functions, f, that satisfy an equation involving derivatives of f.
• Discrete math introduced methods for counting (premutations and combinations).

These courses form the common foundation of engineering and math majors.
They focus on the application of mathematics to solving problems, including calculating forces, accelerations, and strengths of magnetic fields, as well as describing heat transfer, population dynamics, planetary orbits, and compound interest.

Analysis of Functions of a Real Variable (Real Analysis, for short) is one of the next courses a math student would take.
The course is sometimes called Advanced Calculus or Theory of Calculus.
The "Real" means that the function arguments and values will be real numbers (as opposed to complex numbers).
Calculus is sometimes described as consisting of limits, derivatives, integrals.
So "advanced calculus" suggests fancier techniques for calculating limits, derivatives and integrals.
But it might be more accurate to call the class "Advanced Approach to Calculus."
In this course, students learn to prove many of the results they learned in Calculus.

But why are differentiation and integration worth analyzing?

• At a surface level, this course can be seen as giving a deeper understanding of Calculus.
  
  
  • You can safely use Calculus only when you really understand it.
    For example: When is it true that
    limit_n→∞ [∫f_n(x)dx] = ∫ [limit_n→∞f_n(x)] dx ?
  
  
  • You really understand Calculus only when you can prove all the results.


• It may be that a deeper understanding of differentiation and integration is not so intrinsically important.
  Perhaps the main motivation of this course is that studying the proofs of known results (from Calculus) prepares the student to prove new mathematical results (i.e. to be a mathematician).
  The fact that the results are already familiar allows the student to spend less time understanding what is being proven and more time on understanding the how it is being proven.

Course Outline

Earlier math courses have a similar format: introduce a new type of problem and present a technique for solving it.
Real analysis can be seen as a systematic review and explanation of old ideas (numbers, limits, sets, functions, etc.), with the goal of getting to differentiation and integration.

1. Because the functions of interest take real numbers as inputs and produce real numbers as outputs, the course starts with making sure we understand what the real numbers are.
   This turns out to be more complicated than one might expect.
   
   
   1. Most people will remember from their K-12 education:
      
      
      • Integers (..., -3, -2, -1, 0, 1, 2, 3, ...) can be constructed from counting numbers (1, 2, 3, ...) by adding negative signs and the number zero.
      
      
      • Rational number can be constructed by taking ratios of integers.
      
      
      • The real numbers include the rational numbers, plus irrational numbers like sqrt(2), e, and pi.
      
      
      • Complex numbers can be constructed from real numbers by taking two real numbers, a and b, and multiplying one by i = sqrt(-1), giving a + bi.
   
   
   2. But that thirst step was a little vague.
      What are the real (irrational) numbers?
   
   
   • One approach is not to define what the real numbers are and just to list axioms describing the properties (commutative, distributive, associative, etc.) of real numbers that the student will need.
     (a+b = b+a, a*(b+c) = ab + ac, etc.)
     
     Since these are properties we use automatically, there will be no surprises except that so much effort is spent spelling out the obvious.
     
     Though the properties will seem obvious, they are worth reviewing.
     Students of linear algebra will already have examples of multiplication that is not commutative (argument order matters in vector cross products and matrix multiplication).
   
   
   • Another approach is
     
     
     1. Construct the real numbers from rational numbers using Dedekind cuts
     2. Define the arithmetic operators on the cuts
     3. Derive all the properties that were just assumed in the first approach
     
     When I took this class, the first day started with a 10 page handout titled, "What are the real numbers?"
     It was a little shocking to learn that I had been using real numbers for years and didn't know what they were.
     If you see this approach, the axiomatic approach may start looking more attractive.
     
     This approach offers the comfort that the real numbers can be seen as constructable from rationals (just as the rational numbers can be constructed from integers and complex numbers can be constructed from real numbers), rather than being something of a completely different sort.
   
   


2. The next topics are limits and convergence because derivatives, integrals and function properties like continuity are all defined in terms of limits.
   In a Calculus class, limits are often computed using an intuitive approach like, “what happens when x gets large?”
   
   An analysis class uses a more formal definition of convergence
   
   

   a_n converges to limit, L ⇔ ∀ 𝜀>0, ∃ N ∈ ℕ s.t. n≥N ⇒ |a_n - L|< 𝜀

   
   and convergence is proven by showing that such an N always exists.
   
   There will be some questions like, “What is the limit?” but more questions like “Prove that the limit is L” or even more abstract questions like, “Under what conditions do we know there is a limit?”


3. A discussion of sets and their properties (closed, open, compact) may seem like a sudden shift in topic.
   But later, these set properties will be needed to characterize the domains and ranges of functions.


4. With numbers, limits, and sets, we can construct functions and analyze their properties (continuity, uniform continuity, monotonicity, boundedness).
   
   Instead of being given e^3x = 3 and solving for x, focus will be shifted to questions about under what conditions we can show that f(x) = 3 for some x (without needing to know which x, or even which f).


5. With limits, sets, functions and function properties, we can finally define derivatives and integrals.
   Function composition is revisited so we can derive the chain rule.

That is a lot of work to prove that it is valid to do what the student has already been doing for years.
But hopefully, the course outline above will help students stay oriented during the class.

Why Real Analysis Requires a Different Approach Than Calculus (Linear Algebra, Differential Equations, etc.)

In previous courses, students learn techniques for solving problems. The focus is on how to apply the technique, not on how to prove that the technique is valid. This means that students often skim the textbook for theorems to see which one has examples that look closest to a given homework problem. Students can learn to apply theorems and techniques without understanding, or even reading, the proofs.

In a real Analysis course, students must demonstrate the techniques of proving things. The proofs of theorems and lemmas in the texbook illustrate the proof techniques that the student is expected to learn. So the proofs cannot be skipped - they are the focus of the course. This requires a different approach to the textbook.

1. Definitions must be understood and memorized.
   It is surprising how many student skip this.
   But without understanding the vocabulary, it is not possible to understand the assignments, much less find the solutions.


2. Every example proof should be studied until the student can reproduce it.
   And it should not be rote memorization.
   The student should know why each step had to be what it was.


3. The textbook should be read carefully and in order.
   
   It is reasonable to assume that the author designed the book to be absorbed as easily as possible. That means the order matters.
   So every definition, lemma, theorem, and example are there for a reason - to teach you.
   On each page, the author may assume you have the material from all previous pages as context.
   
   Skimming without understanding is reading without the context the author intended.
   And since homework exercises may not assigend in the order that the associated material appears in the book, trying to read only the parts of the book related to the exercises is trying to learn the material in an unintended (harder to absorb) order.

So I recommend reading the textbook in order, before attempting homework exercises.

Even if you read the book in order and understand the proofs, there is one more complication. Students are shown many proofs and are expected to learn how to come up with their own proofs. The problem is that the way proofs are written does not convey how they were written. So it can seem like being expected to learn brainstorming and editing from looking at final written documents. It may work for some, but it would be nice if there were an easier way.

The rest of this page will focus on describing an approach to studying and producing proofs.

Examples of How to Produce Proofs

Main Idea: The way proofs are presented does not show how they were produced.

A common mistake I have seen in correcting students homework is that they finish a proof and keep on going.
It's like they don't know where the proof ends.
They hadn't leaned the basic structure of a proof - what is required, and when it's done.
So a brief review of proofs is in order.

When one thinks of techniques of proof, one often thinks of proofs by induction and proofs using the contrapositive.
But the most basic structures in a real analysis course are proof by definition and proof by theorem.

Techniques of Proof - Proof by Definition

Generic Definition: A sequence, a_n, is said to have property, P, iff condition C is true.
The definition has an "if and only if" (sometimes indicated with "⇔").


Structure of a proof using this defintion in the backward direction:

1. Show that condition, C, is true.
2. Then a_n has property, P, by definition.

Structure of a proof using this defintion in the forward direction:

1. Assume a_n has property, P.
2. Assert condition, C, is true by definition.
3. Use condition, C, to prove something else.

These two patterns are illustrated in the next two example proofs.

Example 1: Known Sequence and Known Limit

Using the Definition of Convergence

a_n converges to limit, L ⇔ ∀ 𝜀 > 0, ∃ N ∈ ℕ s.t. n ≥ N ⇒ |a_n - L| < 𝜀

show that a_n = 1 + 1/n² converges to the limit, L=1.

Notes:

1. The question is not "What is the limit of a_n?"
   From previous classes we probably remember that as n gets large, a/n² approaches zero, so a_n approaches 1 + 0 = 1.
   


2. This is typical of a real analysis class.
   We already know how to calculate the limit.
   This class is about how to prove that our answer is correct.
   


3. Proving things we know to be true allows us to focus on the thinking process for proofs, rather than the result being proven.
   But it can seem pointless to spend so much effort proving what is known (and possibly obvious).

Click Here for a Walkthrough of Writing the Proof

Key Takeaways:

1. The order in which the proof is presented is not the order in which the proof is constructed.
   Construction: Start with the conditions from the definition, then find N(𝜀) at the end.
   Presentation: N(𝜀) is given at the beginning, and then we show the conditions of the defintion are satisfied.
   If one reads only the proofs, key parts (like 1 + ceiling(sqrt(1/𝜀))) seem to appear out of thin air.
   This is part of what makes studying proofs in a book difficult.


2. The proof demonstrates that for any 𝜀>0, there is an integer N with the required properties.
   The existence of an appropriate N was established by writing N explicitly as a function of 𝜀.
   Other proofs might might establish that such an N exists without knowing the functional form.


3. Multiple basic properties of the real numbers may be needed to justify even what look like trivial arithmetic manipulations.

Example 2: Unknown Sequences and Unknown Limits

Using the Definition of Convergence

a_n converges to limit, L ⇔ ∀ 𝜀 > 0, ∃ N ∈ ℕ s.t. n ≥ N ⇒ |a_n - L| < 𝜀

show that if a_n converges to limit, A and b_n converges to limit, B,
then a_n + b_n converges to A + B.

Note:

1. Unlike the previous example, we do not know the form of the sequences or the limits.
   So we will not be able to write N as an explicit function of 𝜀.

Click Here for a Walkthrough of Writing the Proof

Key Takeaways:

1. Again, the order in which the proof is presented is not the order in which the proof is constructed.
   Construction: Find 𝜀_a and 𝜀_b at the end.
   Presentation: 𝜀_a and 𝜀_b are given at the beginning.
   If one reads only the proofs, key parts (𝜀_a = 𝜀_b = 𝜀 / 2 and N = max(N_a(𝜀_a), N_b(𝜀_b))) seem to appear out of thin air.
   The way to study the proof is to try the proof on your own and figure out the thought process that motivates the key parts.


2. This is an example of proving the existence of N without knowing what N is.


3. Patterns:
   Much of the work in proofs is applying definitions and arithmentic, but there are some patterns that are repeated.
   This example introduces a common pattern: splitting 𝜀.
   Sometimes you want a sum of n terms to be less than 𝜀.
   Then you should check for conditions that force each term to be less than 𝜀/n.
   


4. Notation:
   In cases with multiple converging sequences, adding a subscript or argument to the 𝜀 and N may reduce confusion (for the student and the grader).
   In this example 𝜀_a is the 𝜀 from the definition of convergence of a_n and N_a(𝜀_a) is the N associated with that 𝜀_a.

Techniques of Proof - Proof by Theorem

Using theorems to prove things is similar to using definitions, except that theorems often have multiple conditions that must be met.
Also, there can be more than one applicable theorem.

The basic structure of the proof is: (1) Show that all conditions are satisfied, (2) apply the theorem to assert the statement to be proven.

Problem: Prove that s_n = ∑_{i=0 to n}1/i! converges.

Click Here for a Walkthrough of Writing the Proof

Key Takeaways:

1. Again, the order in which the proof is presented is not the order in which the proof is constructed.
   Construction: N is found at the end.
   Presentation: N is given at the beginning.
   If one reads only the proof, the key part (N = ceiling(log₂(2/𝜀))) seems to appear out of thin air.
   The way to study the proof is to try the proof on your own and figure out the thought process that motivates the key parts.


2. This page shows an example of how you can prove the same result with two different theorems.


3. Patterns:
   When applying theorems, you must make sure that every condition is met.
   The proof will be a sequence of sub-proofs, one for each property.
   After those, applying the theorem may be just one line.


4. Notation:
   When your proofs get long, you might want to label intermediate results for easier reference later in the proof.
   Using different colors or boxes around key points can make your proof easier to follow.


5. Organizing the theorems:
   This example considered only two theorems.
   But a typical class will have many theorems.
   One way to organize the theorems in you head is to group them by inputs and outputs.
   Inputs are the conditions that must be met (monotonic, bounded, Cauchy).
   Outputs are the conclusions (the squence converges, the sequence converges to limit L, the function is differentiable).

Study Suggestions

Work in Progress

• As mentioned above, read the book (or lecture notes).
  Reading in order will probably make the material as easy as possible to absorb.


• When studying defintions, test what happens when a single property is omitted.
  For example, for convergence of a sequence
  
  

  a_n converges to limit, L ⇔ ∀ 𝜀 > 0, ∃ N ∈ ℕ s.t. n ≥ N ⇒ |a_n - L| < 𝜀

  
  
  • what happens if you change it to
    
    

    a_n converges to limit, L ⇔ ∀ 𝜀 > 0, ∃ n s.t. |a_n - L| < 𝜀

    
    This would include all sequences, using L = a₁.
    
  
  
  • what happens if you change it to
    
    

    a_n converges(?) to limit, L ⇔ ∀ 𝜀 > 0, N ∈ ℕ, ∃ n ≥ N ⇒ |a_n - L| < 𝜀

    
    This would include sequences that violate our common understanding of convergence.
    The osciallting sequence a_n = (-1)ⁿ would satisfy this defintion with L = 1 or (-1).
  
  
  • what happens if you change it to
    
    

    a_n converges(?) to limit, L ⇔ ∀ 𝜀 ≥ 0, ∃ N ∈ ℕ s.t. n ≥ N ⇒ |a_n - L| ≤ 𝜀

    
    This would exclude sequences that meet our common understanding of convergence.
    It requires that the sequence attains the limit value since 𝜀 can be chosen as zero.
  
  These exercises help the student inderstand why the definitions are what they are.
  They also prepare the student for common test questions of the form, "state the definition of ..."
  


• As mentioned in the example proof key takeaways, don't just read proofs start to finish.
  The way to study the proof is to try write the proof on your own and figure out the thought process behind all those N and 𝜀.
  And remember that the order in which a proof is written may not reflect the order in which is was produced.


• When studying theorems, test what happens when a single property is modified or omitted.
  
  For example, there is a theorem that monotonic sequences converge if they are bounded.
  What happens if you leave out one of the conditions (monotonic or bounded)?
  • Can you think of an example of a bounded (but not monotonic) function that does not converge?
    
  • Can you think of an example of a monotonic (but not bounded) function that does not converge?
    

These exercises help the student inderstand why the theorems are what they are.
They also prepare the student for test questions with a (possibly modified) statement of a theorem and
"True or False. If false, give a counter-example."


• When you write proofs, organize your steps so they are easy to follow.
  I think the common (but harder to read) style of proofs in paragraph form may be left over from when authors wrote paper books and printing was expensive.
  Remember that proofs are a kind of communication.
  Make them easy for your audience to absorb.


• Consider typing your math.
  These days, the options for color, and print size are an advantage to doing math in electronic form.
  Also, copy/paste functions reduce the chance of errors when writing a long sequence of manipulations in which each line has only a small change from the previous line. This encourages the student to write out all the details.
  Microsoft Word and Google Docs, both WYSIWYG, are adequate for deriving mathematical results.
  Typesetting tools, like LaTeX, are less convenient for derivation, but great for final presentation. The disadvantage of LaTex is that it probably takes longer to learn than Word or Google Docs.