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?



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


