Representing the set which contains integers each of which is a subset of certain interval(segment?). I've read the book of cyber security  and saw the notation below.
$$  \mathbb{Z_{n}} ~~\leftrightarrow~~ \text{set of integers which satisfy }~ 0 \leq i < n   $$
And what I want to represent is the set of natural integers equal or greater than 1 and less than n+1
$$ ~~~\uparrow  \left( 1 \leq i \leq n \right)  ~$$
Is there some good notation(s)?
Can I write it as $~ \mathbb{N_{n+1}}  ~$ ? which notation is correct?
I know using $~ \mathbb{Z_{n+1}} \setminus \left\{ 0 \right\}   ~$ is one of the ways but seemingly bit complicated for me.

From proofwiki page, a possible notation is $$\mathbb{N}_n^*=\{x \in \mathbb{N}^*: x \le n\}= \{ 1,2, \ldots, n\}$$
$\mathbb{N}_{n+1}$ includes the element $0$ according to the page as well.
Do remember to define it explicitly in your work to avoid confusion of your readers.
