# What are some easy papers in mathematics understood by undergraduates?

Being a junior(3rd.year) undergraduate in mathematics, I would like to learn how serious papers of mathematics do the proofs rigorously as opposed to textbooks and proffessors in lectures and I wish to see this by myself. For this reason , I just want to read at least one paper which I can understand the topic and proofs.

Though the aim of this reading process is to learn the extent of rigorousness in papers , I want also the subject to interest me so that I would learn something. That is why I might choose the topics somewhat related to real analysis since I consider doing Phd on real& functional analysis or probability & stochastic analysis.Hence, subjects could be Real Analysis, Game Theory, Probability or Stochastic Processes.

The thing is that I should be able to understand the paper to avoid throwing the paper away. I am at the level of Apostol's Mathematical Analysis(Point set topology ,metric spaces, differentiation &integration, continuity, uniform convergence, series and sequences of functions etc.) and I took a course on Lebesgue Integration which started Lebesgue Theory by sequence of step functions. I have not taken or read any thing on Stochastic Calculus but I know Applied Probability and Statistics at the level ,say undergraduate engineering student.

With all these in mind, which spesific paper do you suggest me to read?

1) The paper that I am looking for can be any theorem which is usually a subject of an undergraduate real analysis courses or from a textbook on real analysis. To give some examples, Fubini's Theorem, Beppo-Levi's Theorem in Lebesgue Integration or Banach's Fixed Point Theorem, Arzela-Ascoli Theorem, Baire Theorem etc. on Analysis can be candidates .

• This is such a broad topic which clearly has the tendency on getting "down-votes." (I hope not) So what's your field of interest, Mathematical Analysis ? – user399481 Dec 25 '16 at 8:30
• the paper I am looking for can be on any theorem which is a subject of undergraduate mathematical analysis courses. For example Fubini's Theorem on $\mathbb{R}^n$ – Quantes Dec 25 '16 at 8:33
• Right. Couple of ways on top of my head. Check "Google Scholar" option and search on the topic of your interest. Also, go to arxiv.org, choose Mathematics and do the same. – user399481 Dec 25 '16 at 8:35
• If I do that It can go very broad set of papers which I might choose one that I migh not able to understand even a single line of the paper at all. That is why I am looking an answer here. – Quantes Dec 25 '16 at 8:39
• It's not an easy process considering the level you're at. Either you asked some papers from an instructor who knows your level of understanding. Or you do it by yourself. It's up to you. Initially it'll be hard. But you'll get use to it. – user399481 Dec 25 '16 at 8:42

Suggestions:

Source: Mathematics Magazine, vol. 40, 1967, pp. 179-186.

Source: The American Mathematical Monthly, Vol. 78, No. 9 (Nov., 1971), pp. 970-979.

• Thank you! I will certainly look deeper into your suggestion @Pedro and I am open to other suggestions as well. – Quantes Dec 25 '16 at 13:02
• arzela has many generalisations – Max Dec 25 '16 at 14:14

Another Suggestion:

A simple proof that $\pi$ is irrational

It is a one page proof. It is not a proof of a major theorem, and questionably fits into your field though.

• It is a one page but, in my opinion, in order to make it readable for an undergraduate student we have to expand it considerably, as done in this series (or in this text in portuguese). – Pedro Mar 29 '17 at 1:07