Notation for "all integers less than $n$" 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?
 A: $ \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.
A: I would use 
$$ \{ x \in \mathbb Z \mid x < n \} . $$
A: I have seen the notation $[k<n]$ used several times, first in a combinatorics class when I was an undergrad, and then in another place. By induction, it should hold for all authors.
A: Maybe (almost surely) it's not so standard, but sometimes I saw the use of $\bar{n} $ to indicate the set of natural numbers less or equal to $ n $. I suppose there is an analogue notation for integers.
For references: I found this notation in several random pdfs downloaded from Internet, not "ufficial" books (at least book i know). I addee this answer because I found the notation suitable for someone who doesn't want to write "long" formulas :)
