Project

Real Analysis

due: Friday, 25 July 2025

Overview: The Project counts for 10% of your course grade, and is meant to enrich your understanding of the material covered in the lectures. There are multiple options for the project; each options allows you to create mathematical content using a different medium. The project will be graded according to displayed effort and mathematical accuracy.

Instructions: Choose one of the project options by 11:59pm ET on Sunday, 6 July by completing the “Project Choice Assignment” in the Module 5 folder on Canvas. Complete the project and upload your work by 11:59pm ET on Friday, 25 July by completing the “Project Submission Assignment” in the Module 8 folder on Canvas.

Option 1: The Paper. Choose one of the topics below; independently research the topic and write a 3 page “math paper” that includes the following:

1. A brief history of the topic.

2. A mathematically precise definition or construction.

3. Several mathematical facts concerning the topic, formated in terms of propositions and/or theorems.

4. Related mathematical ideas you discover during your research.

Your paper should include at least one proof; more are preferred. You should properly cite any reference sources. I recommend writing your paper using LaTeX, but any document editing software that allows for mathematical expressions is acceptable.

I recommend this option to any student who wants to expand the breadth of their mathematical understanding and study a topic related to, but not directly covered in, the course.

Topics:

1. The construction of the Real Numbers.

2. Cardinality; the different “sizes of infinity”.

3. The Axiom of Choice.

4. The Cantor set.

5. Pathological functions (nowhere-differentiable function, space-filling curves, etc.).

6. The Rearrangement Theorem for conditionally convergent infinite series.

7. Cesaro and Abel summability.

8. Connectedness and Path-connectness.

9. Measure Theory.

10. Sequences and Series of Functions (there is a lot to research here; choose some subset of the facts you find).

11. Fourier Series.

12. Another topic you find interesting and get approved by me.

Option 2: The Lecture. Choose a lecture from the course, or choose parts of several lectures that together form. a coherent train of mathematical thought. Create a virtual lecture, similar to the ones I produced for the course, that discusses the material you choose. Your lecture should be at least 45 minutes in length, in which you introduce at least one definition, provide at least one example, and prove at least one important proposition or theorem. Your video/audio quality need not be excellent, but you should practice your lecture and/or edit your recording so the final product could be used by a classmate in order to learn the material you discuss. You can prepare your visual content using Powerpoint, a virtual whiteboard (as I do), or something similar.

I recommend this option to any student that wants to improve upon their mathematical communication and “develop their mathematical voice”.

Option 3: The Problems. Solve the following three exercises; each guides you through the proof of an important theorem or idea from Real Analysis. Type your solutions using LaTeX; I suggest using Overleaf to do so. Your solutions should leave no mathematical stone unturned - this is your opportunity to demonstrate your proving skills!

Next, choose one of the problems and record a 10-15 minute virtual presentation (using Zoom, Youtube, etc.) in which you discuss your proof. You should mention definitions and theorems from the course that are relevant to the problem and your solutions, and justify the steps of your proof so that a classmate who didn’t choose this project option could easily follow along and understand the details of the proof. You can prepare your visual content using Powerpoint, a virtual whiteboard (as I do), or something similar.

I recommend this option to any student who wants to expand the depth of their mathematical under-standing and study important consequences of the material covered in the course.

1. In this problem, we prove a special case of the Contraction Mapping Theorem; this is arguably the most important result we’ve yet encountered in mathematics. First we prepare for the proof with some preliminary results.

(a) Suppose f is continuous and that the sequence

x, f(x), f(f(x)), f(f(f(x))), ...

converges to l. Prove that l is a fixed point of f, i.e., that f(l) = l.

(b) Show that if c ≠ 1 then

(c) Suppose that |c| < 1. Prove that

(d) Suppose that {xn} is a sequence satisfying |xn − xn+1| ≤ c n for some 0 < c < 1. Prove that {xn} is a Cauchy sequence.

Now we prove the theorem. Argue carefully!

Theorem. Suppose f : R → R is a contraction, i.e., f satisfies

|f(x) − f(y)| ≤ c|x − y|   for all x, y ∈ R

for some c < 1. Then f has a unique fixed point, i.e., there exists exactly one x0 ∈ R such that f(x0) = x0.

(e) Suppose f is a contraction with associated contraction constant c < 1. Prove that f is continu-ous.

(f) Prove uniqueness of a fixed point, i.e., prove that f has at most one fixed point.

(g) Prove existence of a fixed point, i.e., prove that f does have a fixed point. You should consider the sequence

x, f(x), f(f(x)), ...

for an arbitrarily chosen x ∈ R.

2. In this problem we’ll prove that every continuous function is integrable. In order to prove integrability we’ll need a stronger version of continuity.

A function f is said to be uniformly continuous on A if for every ϵ > 0 there exists a δ > 0 such that for all x, y ∈ A if |x − y| < δ then |f(x) − f(y)| < ϵ.

The difference between uniform. continuity and regular continuity is that in uniform. continuity the δ does not depend on x, i.e., the same δ should work in the definition of continuity for all x ∈ A.

(a) Prove that if f is uniformly continuous then f is continuous.

(b) Prove that f(x) = x is uniformly continuous on R.

(c) Prove that f(x) = x2 is uniformly continuous on [0, 1] but not uniformly continuous on the real line R.

(d) Prove that f(x) = x/1 is not uniformly continuous on (0, 1].

(e) Prove that if f : A → R is continuous and A is compact, then f is uniformly continuous on A.

(f) Let f be continuous (and hence uniformly continuous) on [a, b]. Prove that for all ϵ > 0 there exists a partition P of [a, b] such that U(f, P) − L(f, P) < ϵ. Conclude that f is integrable on [a, b].

3. Working with integrable functions is challenging because they need not be continuous. In this exercise we investigate step functions, which are easy to work with and closely approximate integrable functions. They are frequently used in mathematical analysis!

A function s : [a, b] → R is step function if there is a partition P = {t0, ..., tn} of [a, b] such that s is constant on each (ti−1, ti). The values of s at each ti may be arbitrary.

(a) Show that if s1 and s2 are step functions on [a, b], then s1 + s2 is as well.

(b) Prove directly from the definition of step functions and of the integral (i.e., using Proposition 7.20 (2)) that 

(c) Prove that if f is integrable on [a, b], then for any ϵ > 0 there is a step function s1 ≤ f with  and also a step function s2 ≥ f with 

(d) Suppose that for all ϵ > 0 there are step functions s1 ≤ f and s2 ≥ f such that  Prove that f is integrable.

(e) Use the above to provide an alternate proof of Proposition 7.20 (2). Your proof should not involve upper/lower sums.

(f) One application of step functions is that we can approximate integrable functions with contin-uous functions. Prove that if f is integrable on [a, b] then for any ϵ > 0 there are continuous functions g ≤ f ≤ h with  You should use part (c) to get started; if you find the construction of your continuous functions to be challenging, try drawing some pictures to help motivate yourself.

Option 4: The Wild Card. Come up with your own project. If you choose this option, you should get your project idea approved by me by the end of Module 5 so you have time to complete it.