Mathematical usage of "$\dots$" during enumeration, is it ok to be imprecise?

Question

I am guilty of writing things like the following in proofs:

so by lemma 1.2 we have that for $k<n$ all of the integers $k,k+1,k+2,k+3,\dots ,n$ are pompous.

I really like how this looks, the problem is that $k+3$ might be larger than $n$, so I think this is technically wrong.

Is this sort of thing frowned upon? How do you avoid this problem? Is it ok to do this?

Answer 1

It depends on your audience. If your readers are mathematically quite immatrue, such a phrase might cause trouble. But in general, there's only one possible interpretation, so why bother spelling out the other cases?

Answer 2

Note that for your particular example, you could write "all of the integers in $[k,n]$ are pompous" or "all of the integers $i$ with $k \leq i \leq n$ are pompous" if you want to avoid the ellipsis.

In general, though, ellipses like these are fine. An argument that is too informal and misses important details is problematic, but so is one that is excessively rigorous to the point of obfuscating the argument. Striking the right balance is a matter of experience and style. "Let $x_1, x_2, \ldots, x_n$ be pompous integers" when possibly $n\leq 2$ is a case where favoring clarity of expression over perfect rigor is almost always preferable, and requires no special comment. (If it later becomes important that $n>2$, I would explicitly point that out: "for $n>2$, let $x_1,x_2, \ldots,x_n$ be pompous integers," etc.)