Should one state the sharpest version of a Lemma even if only a weaker version is needed?

Question

When writing a paper, it's possible that some auxiliary results hold in more generality or in a stronger version than what's actually needed to prove the main results of the article. And so here comes the question:

Should one state and prove the exact auxiliary result that is used, or should one sharpen it to its best possible version?

I can think of pros and cons of both approaches: Proving better results cannot be a bad thing in itself, but spending time proving a too strong and not-so-interesting Lemma might be distracting and not worth the effort. Even if it's not so hard to improve the Lemma it might be confusing to the reader to use a weaker version of what's stated.

Example:

Suppose I need to use a Lemma of the form:

For every $\varepsilon>0$ there exists a sequence $(x_n)_n$ with property $(P)$ such that $|x_n|<\varepsilon$ for all $n\in\mathbb{N}$.

However, looking at the proof of this Lemma I (and most likely the referee and the reader) noticed that slightly changing the proof a stronger version holds:

For every sequence of positive numbers $(\varepsilon_n)_n$ there exists a sequence $(x_n)_n$ with property $(P)$ such that $|x_n|<\varepsilon_n$ for all $n\in\mathbb{N}$.

Which version should I include if I only need the first (and weaker) statement?

Answer 1

It depends on context. Here are some relevant considerations: How much more difficult is the stronger Lemma to prove? If the proof is nearly identical, then stating the strongest one may be a good idea. But it may not be helpful to spend a lot of time on making a minor Lemma slightly stronger if it takes a lot of effort and distracts from the main exposition. One thing I've occasionally done is included two versions of a Lemma, and explicitly called a stronger one a "Proposition" and make clear that it is a separate question of just how far we can push the Lemma. (Which may be an interesting question in its own right.)

One thing to also keep in mind is that at least stating the strongest version may help others in two other ways. First, if they extend, generalize, or improve your result, having a stronger version of the Lemma may also be helpful. Second, if someone is trying to understand what the limiting steps are in your main result (e.g. why does this only apply to p-groups but not nilpotent groups, or why can some general inequality not be strengthened, etc.) then having a Lemma which is stronger than you need it can help understand that that Lemma is not the where whatever obstruction there is to making the result stronger.

Another consideration is that a stronger Lemma may benefit you later. If you come back to a problem years later, you might not remember the stronger version, or might remember it but might not remember the proof. So if you decide not to include the proof in the final paper, it may be a good idea to keep a written up version in your own copy, or possibly just commented out in the LaTeX.

Answer 2

In each individual case, it is a question of opinion and of the author's judgement about the importance of the lemma.

There is no logical distinction between a lemma and a theorem.

Could Hermann Amandus Schwarz guess that 150 years later a large part of his fame would be based on the "Schwarz Lemma" which he did not care to state and prove in full generality? (The modern statement and proof are due to Carathéodory.)

Or Rolf Nevanlinna, whose Lemma on the Logarithmic Derivative is probably more important than the Second Fundamental Theorem in the same paper?

A 2010 Fields medal was awarded to Ngô Bảo Châu for the proof of a Lemma.

See also the List of Lemmas in Wikipedia.

Answer 3

In general, I think that it is a good idea to include the stronger version of the lemma. It highlights your ability to think more deeply about the problem, and it shows that you are aware of the generality of the result. It also serves as a good reference for future readers who may need the stronger statement.

As to whether it is worth the effort, I think that depends on the difficulty of the proof. If it is relatively straightforward to prove the stronger version, then it is probably worth doing. If it requires a significant amount of effort to prove the stronger version, then it may be best to just include the version you need.

Answer 4

I see some good arguments for using the stronger lemma among the answers, so let me provide an argument for the weaker one. Typically, stronger lemmas require stronger assumptions -- and conversely, weaker ones let you get away with fewer. I think it's good practice to prove your result with the simplest tools possible, because these are typically the easiest to generalize. This principle suggests using the weakest lemma that gives you the needed result -- since it's probably also simpler and easier to pick apart and tinker with than the more complicated lemma.

An added (not entirely unrelated) advantage of using simple tools when possible is accessibility. You presumably want your result to be understandable by a maximally broad audience, and its reliance on advanced tools may unnecessarily narrow the scope of the reach.