# What is the proper notation for a sequence (or function) which starts at zero and has the upper limit of 1 that is never reached?

If we have a function

f(a, b) = (a-1)/b

where a and b are integers and the values of a range from 1 to b,

what would be the proper way to use notation to describe when the sequence starts and ends? Its lowest value is 0, when a = 1. Its highest value is hypothetically 1, although this is never reached, but you get closer to 1 as you increase b.

You can write $$\sup_{\textstyle {1\leqslant a\leqslant b \atop b\in\Bbb N}} f(a,b) = 1\quad\text{ and }\quad f(a,b) < 1$$
or using the domain $$f$$: $$D=\{(a,b)\in\Bbb N\times\Bbb N \mid 1\leqslant a\leqslant b \}$$
$$\sup_{(a,b)\in D} f(a,b) = 1$$
This doesn't however encode the fact that you need $$a,b\to\infty$$ in order to get arbitrarily close to the supremum. Notice that this condition is only necessary but not sufficient; take for example $$b=2a$$ so that both $$a$$ and $$b$$ tend to infinity but $$f(a,b)\to 1/2$$.