[Math] Notation for “all integers less than $n$”

Question

Is there a short mathematical notation for all integers less than $n$ where $n$ itself is some integer? The only thing that comes to mind is
$$\mathbb{Z} \cap (-\infty, n),$$
But this is pretty ugly and involves reals. Is there a better way to do this symbolically, or should I just describe such set with words?

Answer 1

$ \mathbb{Z}^{< n} $ is sometimes used (I personally like the notation $ \mathbb{Z}_{< n} $ more as I like to reserve the right upper corner for powers, but I don't think I have seen others use it). The $< n$ in the top right can look a little bit weird, because there is no variable on the left side.

Another option is $ k \in \mathbb{Z} $, with $ k < n$. It is also pretty common to just write "for all integers less than $n$" as you did. I think these two options are the most commonly used.

Also, you could write 'for $k \in \{ j \in \mathbb{Z} : j < n \}$', but I don't think many authors would do so. I'm sure there are some other possibilities, but these are the most common ones I can think of.