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](https://academia.stackexchange.com/questions/164114/how-to-structure-a-proof-by-induction-in-a-maths-research-paper?noredirect=1#comment442141_164114) 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?



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


