How to structure a proof by induction in a maths research paper?

I am 16 years old at the time of writing (so I have no supervisors to seek advice from) and I have written a mathematics research paper, which I plan on submitting to a journal for publication. I asked an identical question on Academia.SE and I was advised to ask the question here.

For a couple of the assertions that I make, I use proofs by induction. Now, in school we're encouraged to write proofs by induction in the following (rigid) format:

Base case:...

Assumption(s): ….

Inductive step: ….

Conclusion: ….

I have noticed that no research articles that I have seen have written proofs by induction using this sort of format. The authors usually make it flow much more smoothly, eg 'For the base case, the result is trivial. Now assume the result holds for some $$n=k$$, so that …. Now consider the expression for $$n=k+1$$ … and by the inductive hypothesis this equals … hence the result is true by mathematical induction.'

So, is it good practice to write proofs by induction in the pretty rigid structure I first outlined or is it ok/better to write the proofs more naturally so that it flows better?

• I don't think that this is an MO question, but I'm not sure where else it goes, and it would be a shame to see it go unanswered, so I do not vote to close. \\ I think that the answer is that you should write proofs to the standard of the journal to which you intend to submit. I agree that most published proofs do not follow the rigid proof structure that is learned in an introductory proof class. \\ (By the way, congratulations on your accomplishment!) Commented Mar 19, 2021 at 14:19
• Math Stack Exchange link is probably a good place for this question. Good luck! Commented Mar 19, 2021 at 14:25
• @A-LevelStudent The answers you received on Academia SE all seem very thorough -- is there a specific part of your question that you don't think is answered by the answers you received there? Commented Mar 19, 2021 at 15:05
• If we ignore the first sentence, I think this would become a question that might be applicable for professionals as well. By the way, there are already some good advices on this site about how to write proofs in general, and how to teach undergraduates to write proofs Commented Mar 20, 2021 at 6:54
• @LSpice Based on what's written in the help center this question seems to be on topic (mathematical questions related to current research) if one adopts the viewpoint that proof writing/communication is part of mathematics. (I don't know what fraction of MO users are of this opinion, however.) Commented Mar 20, 2021 at 17:13

Writing a proof for school is very different from writing a proof for a research paper. Perhaps the most important distinction is that the audiences are completely different. In school, your audience is your instructor, whose job is to assess your ability to learn and apply a principle. The audience of a research article is the professional mathematical community, where favorable viewing of your work hinges on novelty of ideas, correctness, readability, and possibly elegance, not rigid adherence to one person's notion of how to organize thoughts. With that in mind, I know I would prefer to read a proof with a nice natural flow instead of one that is written in rigid adherence to one specific instructor's preferences.

When you finish writing your paper, I recommend that you send your paper to a professional researcher with whom you have a good working relationship, someone who can give you candid, meaningful, and constructive feedback. As you go about writing your paper, I recommend reading as many papers in professional journals as you can so that you get a sense for what good writing looks like. This second bit of advice is tricky without knowing what area(s) of research interest you. So perhaps the professional researcher with whom you have a good working relationship might direct you to some examples of quality writing.

• A good approximation of "reading as many papers in professional journals as you can" would be "read, and then make sure you cite, the relevant literature." Then write proofs that would be in the same sort of vein (filling in any additional details in the margins, or in LaTeX comments). Commented Mar 19, 2021 at 17:18
• @A-LevelStudent If you live near a college/university, perhaps you could arrange a meeting with a professor close to your area of research. If that is not applicable, you could try sending an email to one or more of the professors that you cite in your paper. An internet search might indicate whether a professor you might consider contacting is involved in mentoring undergraduates, which might provide some indication of whether they'd be willing to engage with someone who is not yet a full-fledged professional. Commented Mar 19, 2021 at 19:07
• I would like to second the advice about getting a professional mathematician to look at the write-up. Even among Ph.D. students, it is extremely rare for their first paper to be written acceptably well to be considered for publication before their advisor makes a pass or two over it with a red pen. Commented Mar 19, 2021 at 20:07
• Indeed, if someone would even bother mentioning it, one would write "$\sum_{k=1}^n(2k-1)=n^2$ follows readily by induction" instead of even mentioning a base case $n=0$, an induction hypothesis, or an inductive step. Commented Mar 20, 2021 at 11:36
• @HagenvonEitzen If I were to write that in a paper, I would say "by induction on $n$" rather than just "by induction." In a research paper, many things are elided because the reader is expected to be able to fill them in, but one thing that any proof by induction has to be clear about is what variable the induction is proceeding on. In your example, there is only one variable, so there is no danger of confusion, but the habit of specifying the variable has been so ingrained in me that I would say "induction on $n$" anyway. Commented Mar 20, 2021 at 13:14

Welcome to math overflow!

There is no need to list up the names of the steps of the induction (Base case, Inductive step etc.) if it clear from context and/or obvious what you're doing. This will likely only make the argument take up more physical space on the page than strictly necessary. It should however be clear that you are doing an argument by induction, what the base case is, and how the induction step is performed.

If a step of the argument is a little bit difficult to comprehend mentally, then I would say it's no loss in making it explicitly clear why the argument holds, without drowning in details, of course.

I had a teacher once that used the term "Weierstrass-rigorous" for certain arguments / persons. This refers to the ideal standard that your proofs should be logically consistent (not contradict them selves) and "water-proof" , in other words correct, as others have stated. When attempting to confine to such a standard, it often happens that there are steps used in a proof that actually assume more than they seem. For example, something as innocent as

$$"\text{Let} \hspace{2mm}(x_n)_{n=1}^{\infty}\hspace{2mm} \text{be a sequence of elements in} \hspace{2mm} X",$$

may actually require the assumption of the Axiom Of Choice to obtain the sequence.

The proof should also be readable. This means, among other things, that all symbols, functions, etc. that you utilize throughout the proof are defined ether a priori, or on the fly. It is also a bonus if the proof follows a natural progression, if this is possible.

In summary: Too rigorous is better than to little rigorous, and write it in away that makes it readable (or perhaps also enjoyable). And don’t forget to "get to the point" in the proof.

[Ps. Writing style is very much a personal preference, as long as it is rigorous.]

• Thank you for your kind and informative answer! It's great, and should be very helpful later on in my mathematical career. Commented Jul 8, 2021 at 12:01