Let the following set, $$S_n = \left \{0, \frac 1 n, \frac 2 n, \dots, \frac {n-1}{n}\right\}, \quad n \in \mathbb N_+.$$
I read that if
$$n\to \infty \implies S_n \to [0,1]. \tag{1}$$
Namely, that as we "let $n$ go to infinity", the set $S_n$ becomes the interval $[0,1]$ of real numbers.
The only intuition I can think of is that the maximum element of $S_n$ is bounded so we are squeezing more and more numbers in a finite interval... any hint on how I could go about proving (or disproving) assertion $(1)$ formally?