How to structure a proof by induction in a maths research paper?
As you probably have observed by now, by reading articles, people generally do not write steps down explicitly when those steps are considered &quot;obvious.&quot; This trend is not special to proofs by induction.
In fact, this is true at any level, but what is considered &quot;obvious&quot; to an audience of people holding a PhD or with significant progress towards a PhD is different from what is &quot;obvious&quot; to someone taking first year calculus.
Think about it this way: people reading research articles know math quite well already. What they are interested in is what is new in your paper; demonstrating &quot;obvious&quot; facts is just a distraction. And, if you are at the stage where you will read several papers in a week, you generally want those papers to get to the point.
For people who are learning math, it's entirely different: you're probably only reading 1 textbook at a time, and the whole thing is new to you. So, you want everything explained in lots of detail because you've never seen it before. But again, if you're reading a calculus book, you probably don't need an explanation on solving a quadratic; you already know that. If you're learning induction, you want it spelled out; if you've seen proofs by induction ten thousand times, you don't need it spelled out for you.
Since you are a bit new to this, I would like to give some more examples about how proofs by induction are typically structured in upper undergraduate/graduate/research levels. Now, when it comes to proofs by induction, there are generally two categories: proofs where the induction step is &quot;obvious&quot; and proofs where the base-case is &quot;obvious.&quot; Sometimes both are easy, but it's very rare in my experience to find a proof by induction where both the base case and the induction step require a lot of work to prove.
I suppose on the rare instance where both the base case and the induction step are non-obvious, I might split the proof into two lemmas or possibly I would call these &quot;claims.&quot; In every other case, I write statements such as &quot;the base case is trivial&quot; or &quot;the general case (the induction step) follows by induction.&quot;
I would also point out that there are even times where a proof will &quot;prove&quot; something by induction without the author ever writing that induction is being used. For instance:

Let f(n + 1) = 2f(n) and f(0) = 1. Then f(n) = 21 f(n - 1) = 22 f(n - 2) = . . . = 2k f(n - k) = . . . = 2n.

Induction is being used implicitly within the ellipses, but it hasn't been written explicitly so. I, personally, like to write &quot;by induction&quot; at the end of such a statement, but not everyone does.
I think your intuition of generally following how other people write mathematics is appropriate. You can deviate slightly, sure, but it's also good to understand why things are written the way they are and so it is great that you are asking this question.