Teaching Real Analysis—an active approach

Session for 2014-2015Project NExT Fellows

Mathfest, 2013Portland, OR

Presenter:

Carol S. SchumacherKenyon College

Real Analysis or Advanced Calculus?

Real Analysis is the branch of mathematics that allows us to describe limiting processes precisely.

It is crucial that our students be able to make direct connections between the ideas they are studying in their real analysis course and the intuition they developed about limiting processes in their calculus courses.

At the same time, it is important that the course not be simply a re-tread of calculus that appears to do no more than “cross t’s and dot i’s.”

Definition: an L means that for every > 0,

there exists N such that for all n > N, d(an , L) < .

First Definitions

Given any tolerance

there is some fixed position

beyond which

an is within that tolerance of L

In beginning real analysis, we typically begin with sequence convergence.

First Definitions

Definition: an L means that for every > 0, there exists N such that for all n > N, d(an , L) < .

In beginning real analysis, we typically begin with sequence convergence:

Intuition: the sequence (an ) converges to a limit L provided that, as we go farther and farther out in the sequence, the terms of the sequence get closer and closer to L.

We start where the students are

Why isn’t this a definition (in the

mathematical sense)?

Intuition: the sequence (an ) converges to a limit L provided that, as we go farther and farther out in the sequence, the terms of the sequence get closer and closer to L.

Intuition First

Intuition: the sequence (an ) converges to a limit L provided that, as we go farther and farther out in the sequence, the terms of the sequence get closer and closer to L.

Intuition First

The key sticking points are the phrases “farther and farther out” and “closer and closer.” Any mathematically sound definition requires a rigorous understanding of what these phrases mean and how they fit together to give us the behavior that we want.

Intuition: the sequence (an ) converges to a limit L provided that, as we go farther and farther out in the sequence, the terms of the sequence get closer and closer to L.

Intuition First

The key sticking points are the phrases “farther and farther out” and “closer and closer.” Any mathematically sound definition requires a rigorous understanding of what these phrases mean and how they fit together to give us the behavior that we want.

It took people like Gauss and Cauchy and Riemann and

Weierstrass most of a century to get a handle on this; we

shouldn’t be surprised if our students don’t pick it up

immediately.

Definition: an L means that for every > 0, there exists N such that for all n > N, d(an , L) < .

First Definitions

Back to the mathematical definition of sequence convergence.

• an L means that for all > 0 there exists n ℕ such that d(an , L) < .

• an L means that for all > 0 there exists N ℕ such that for some n > N, d(an , L) < .

• an L means that for all N ℕ, there exists > 0 such that for all n > N, d(an , L) < .

• an L means that for all N ℕ and for all > 0, there exists n > N such that d(an , L) < .

Don’t just stand there!Do something.

Students are asked to think of these as “alternatives” to the definition. Then they are challenged to come up with examples of real number sequences and limits that satisfy the “alternate” definitions but for which an L is false.

• an L means that for all > 0 there exists n ℕ such that d(an , L) < .• an L means that for all > 0 there exists N ℕ such that for some n > N, d(an , L) < .• an L means that for all N ℕ, there exists

> 0 such that for all n > N, d(an , L) < .• an L means that for all N ℕ and for all >

0, there exists n > N such that d(an , L) < .

Don’t just stand there!Do something.

Students are asked to think of these as “alternatives” to the definition. Then they are challenged to come up with examples of real number sequences and limits that satisfy the “alternate” definitions but for which an L is false.

First Activity!

Second “epsilonics” definitionYour students have been thinking about

sequence convergence for a while now and they are beginning to get the hang of this new way of thinking. You are ready to tackle continuity.

How do you start with your students’ previous understanding of continuity (from calculus) and end up with the standard - definition of continuity?

Second “epsilonics” definitionYour students have been thinking about

sequence convergence for a while now and they are beginning to get the hang of this new way of thinking. You are ready to tackle continuity.

How do you start with your students’ previous understanding of continuity (from calculus) and end up with the standard - definition of continuity?

Second Activity!

“That’s obvious.”To a mathematician it means “this can easily be deduced from previously established facts.”

Many of my students will say that something they already “know” is “obvious.”

For instance, they will readily agree that it is “obvious” that the sequence 1, 0, 1, 0, 1, 0, . . . fails to converge.

We must be sensitive to some students’ (natural) reaction that it is a waste of time to put any work into proving such a thing.

• First: people don’t begin by proving deep theorems. They have to start by proving straightforward facts.

•Second: this is a sort of ‘test’ of the definition. It is so fundamental, that if the definition did not allow us to prove it, we would have to change the definition.

I Stipulate Two Things

• First: people don’t begin by proving deep theorems. They have to start by proving straightforward facts.

•Second: this is a sort of ‘test’ of the definition. It is so fundamental, that if the definition did not allow us to prove it, we would have to change the definition.

I Stipulate Two Things

Third Activity!

(an ) converges to L if

for every > 0, there exists N such that

for all n > N, d(an , L) < .

(an ) fails to converge provided that

for all L it is not true that “for every > 0, there

exists N such that for all n > N, d(an , L) < .”

Negating Statements

(an ) converges to L if

for every > 0, there exists N such that

for all n > N, d(an , L) < .

(an ) fails to converge provided that

such that d(an , L) .there exists > 0 such that for all N

there exists n > N

for all L

Thinking like an AnalystWe have skills and practices that we use

when we think like analysts.

We hold presuppositions and assumptions that are unlikely to be shared by a student who is new to real analysis.

We know where to focus of our attention and what can be safely ignored.

Thinking like an AnalystWe have skills and practices that we use

when we think like analysts.

We hold presuppositions and assumptions that are unlikely to be shared by a student who is new to real analysis.

We know where to focus of our attention and what can be safely ignored.

How do we get our students to

think like analysts?

Great Versatility is Required•Students must be able to understand and interpret the meaning of statements involving stacked quantifiers•They must be able to prove a theorem in which they establish the truth of a statement involving stacked quantifiers.•Students must be able to use a hypothesis that involves stacked quantifiers.•Students must be able to negate a statement involving stacked quantifiers and then prove it or use it in a theorem.

And these are all different skills that have to be learned!

Great Versatility is Required•Students must be able to understand and interpret the meaning of statements involving stacked quantifiers•They must be able to prove a theorem in which they establish the truth of a statement involving stacked quantifiers.•Students must be able to use a hypothesis that involves stacked quantifiers.•Students must be able to negate a statement involving stacked quantifiers and then prove it or use it in a theorem.

And these are all different skills that have to be learned!

Note: statements that begin with “there exists > 0 such that for all . . .” are handled differently than

statements that start with “for all > 0, there exists . . .”

Great Versatility is Required•Students must be able to understand and interpret the meaning of statements involving stacked quantifiers•They must be able to prove a theorem in which they establish the truth of a statement involving stacked quantifiers.•Students must be able to use a hypothesis that involves stacked quantifiers.•Students must be able to negate a statement involving stacked quantifier and then prove it or use it in a theorem.

And these are all different skills that have to be learned!

I am not suggesting that we categorize all these things for our students, but if we aren’t aware of the differences, we

can’t foresee the myriad ways in which our students can get

in trouble.

Some standard problems

Fourth Activity!

“Epsilonics”

“Organizing principle” for final proof

Spare and stylized

if for every > 0, there exists > 0

such that if , then .

2

23

9lim

1 10x

x

x

0 3x

2

2

9

1 10

x

x

= Kabuki dance

if for every > 0, there exists > 0

such that if , then .

2

23

9lim

1 10x

x

x

3x

2

2

9

1 10

x

x

o Existence theorems Start the proof by thinking about what you want

to prove rather than what you are assuming. You rig the game so you are guaranteed to win.

o The desired conclusion is the “organizing principle” for the write-up.

“Epsilonics”---Some general principles

“Organizing principle” for proof

if for every > 0, there exists > 0

such that if , then .

2

23

9lim

1 10x

x

x

3x

2

2

9

1 10

x

x

o Existence theorems Start the proof by thinking about what you want

to prove rather than what you are assuming. You rig the game so you are guaranteed to win.

o The desired conclusion is the “organizing principle” for the write-up.

“Epsilonics”---Some general principles

“Organizing principle” for proof

Proofs are written backwards!

“Epsilonics” proofsGeneral rules of thumb

To get started, calculate the quantity that you want to make “small.” Must find a relationship between it and the quantity (or quantities) that you know to be “small.”

2

2 2

2

9 ( 3)( 3)

1 10 10(1 )

13 3

10(1 )

x x x

x x

x xx

“Epsilonics” proofsGeneral rules of thumb

To get started, calculate the quantity that you want to make “small.” Must find a relationship between it and the quantity (or quantities) that you know to be “small.”

“The sum of small things is small”---the triangle inequality.

“The product of something small and something bounded is small”

The “multi-task” delta.

“Epsilonics” proofsGeneral rules of thumb

To get started, calculate the quantity that you want to make “small.” Must find a relationship between it and the quantity (or quantities) that you know to be “small.”

2

2

9

1 10

x

x

3x

2

2 2

2

9 ( 3)( 3)

1 10 10(1 )

13 3

10(1 )

x x x

x x

x xx

f fails to be uniformly continuous provided that

f is uniformly continuous if

For all > 0, there exists > 0 such that

if d(x,y) < , then d(f (x), f (y)) < .

For all x, y

such that d(x,y) < and d(f (x), f (y)) .there exists > 0 such that for all > 0

there exist x, y

f fails to be uniformly continuous provided that

f is uniformly continuous if

For all > 0, there exists > 0 such that

if d(x,y) < , then d(f (x), f (y)) < .

For all x, y

such that d(x,y) < and d(f (x), f (y)) .there exists > 0 such that for all > 0

there exist x, y

Activity 5: You are teaching a real analysis class and have just defined continuity. Your students have been assigned the following

Problem: K is a fixed real number, x is a fixed element of the metric space X and f : X is a continuous function. Prove that if f (x) > K, then there exists an open ball about x such that f maps every element of the open ball to some number greater than K.

One of your students comes into your office saying that he has “tried everything” but cannot make any headway on this problem. When you ask him what exactly he has tried, he simply reiterates that he has tried “everything.” What is happening? What do you do?