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?
 A: 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.]
A: 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.
