This week I concluded my proofwriting with a discussion of the exceptionally-tricky “squangular numbers.” I also compiled all of my work this semester into one final document, which I have included below.
Balkus_MTH312_Final_PaperWeek 12 – Analysis Proof Practice
This week, I practiced my analysis skills via several proofs which we later discussed in class. My work and thoughts while tackling these problems are included here.
Proofs for Friday 4-16-21 More_Proofwriting (1)Week 11 – Metric Space Preliminaries
Continuing our investigation of the proof writing process, I have outlined two proofs of important inequalities in the study of metric spaces, one of the last chapters in the textbook which we have not yet covered.
Metric_SpacesWeek 11 – Proof Writing
Over the weekend I corrected some errors in my previous proofs related to the squeeze theorem, and added a few graphs to help visually demonstrate the proof.
Proofwriting (4)Proof Writing for Friday’s Class
On Friday, we plan on reviewing the proof-writing process. Here, I include the proofs that I have prepared for this discussion.
Proofwriting
Week 10 – Riemann Integrability of the Cantor Function
This week, in addition to the assigned reading, I developed a more comprehensive proof of why the Cantor function is Riemann integrable. This involves proving that f_n(x) is uniformly continuous, and that f_n(x) converges uniformly to the Cantor function. The only point of confusion I had was at the end of the proof that f_n(x) converges uniformly, at the end of the section where I prove the induction step by demonstrating that |f_n+p(x) – f_n(x)| < epsilon implies that |f_n+p+1(x) – f_n(x)| < epsilon. I felt like the argument could be made stronger, but I was not exactly sure the best way to go about it.
Cantor_Set (2)The Cantor Function
This week, I made a few additions to my Cantor set and Cantor function discussions, most notably adding a discussion of why the Cantor function converges. I also include a computational notebook that animates the Cantor function and explores the error of the approximation f_n at different values of n.
Cantor_Set (1)Cantor's Function
Week 8 – Concluding Riemann Function, Beginning the Cantor Set
This week, I wrapped up my work on the Riemann function by outlining the proof that the Riemann function is integrable. I completed the work in my notebook before the beginning of class today, but my work wound up very similar to our classwork. Furthermore, I worked on the exercises relating to the Cantor set. These include computational demonstrations for discovering whether a value is in the Cantor set, which are included in the following Mathematica Notebook pdf. It also includes a proof that the Cantor set is uncountable infinite, and that it contains no intervals.
Riemann_Function (2) Cantor Set - Base3 Representations Cantor_Set
Week 7 – Jumps of the Riemann Function
Over the weekend, I worked out a proof regarding the size of the jumps at the discontinuities on the Riemann function based on the work that I completed last week.
Riemann_Function (1)Week 6 Part 2 – The Riemann Function
After this class, we investigated the Riemann function further, and I completed some exercises related to proving where its discontinuities occur.
Riemann_Function