Is it better to prove a generalization of a theorem before presenting specific cases?

Question

In (mostly) mathematical publications, when proving a theorem that can be generalized to a wider range of parameters, is it generally considered better practice to:

• Present a special case of a theorem first, and then prove it in a general context, or
• Prove the general case and then move on to present one (or more) special cases?

The first approach makes understanding the theorem easier, and makes the paper quicker to read. The second one looks more "rigorous", but is much more difficult to understand for more complex theorems and formulae.

What is generally considered the better practice?

Answer 1

There is no "general" answer here. It always depends on circumstances:

• Is the general proof really hard and difficult to understand?
• Is the general proof of interest, or does only the special case have applications (yet)?
• Can the general proof be done in the same way as the special case, just with uglier notation, more indices, etc.?
• Is the generalization natural or does it require work to actually show that the special case does indeed follow from the general one?
• Is the special case the only application of the general theorem, or are there other corollaries of interest?
• ...

This is just a small part of the questions to consider, depending on what the theorem and the proof looks like. In some cases, it might be good to use the special case to motivate the generalization, especially if this is a new idea and you are the first to look at this general case.

On the other hand, people want to see the benefits of the generalized version, so it would be best to get results from it that go beyond just showing a special case.

I would suggest leaving the choice of which proof to read to the reader. So if you have a really easy proof for the special (and most interesting) case and the general proof is really hard, you should give the special case first and let the reader decide if he wants to skip the general one or not. I sometimes see phrases like "readers only interested in the case ... can safely skip to section X". Of course this has to be properly organized, such that section X is still readable even if the general proof got skipped. In this way, it is easy to see and understand the basic concepts of your paper and how you showed the special case, and there is always the opportunity to come back later to study the general one (or to give it to a student for a bachelor thesis...). Note, however, that this is only my personal opinion and approach, this is not a general consensus.