Notation for integer between two values

Notationally, to write $i$ is an integer within a given interval, you could write several different things:

$$i\in\Bbb Z:i\in[1,N]\tag 1$$

$$i\in\Bbb Z:1\le i\le N\tag 2$$

$$i\in\Bbb Z\cap[1,N]\tag 3$$

where each is read as follows:

$(1)$ "$i$ is an integer such that $i$ is within the interval $1,N$"

$(2)$ "$i$ is an integer such that $1$ is less than or equal to $i$ is less than or equal to $N$"

$(3)$ "$i$ is an element of the set intersection of the integers with the interval $1,N$"

Each would be considered a valid representation, and each may be considered more appropriate for given circumstances or writing styles than the others.

As mentioned elsewhere, it is also common to write $i=1,2,\dots,N$ especially when $i$ is an indexing element.

It is very common in mathematics to write simply $i=1,\ldots,n$.

It is almost an universal truth that $i,j,k$ are natural numbers when we write like above.