My textbook begin with a chapter dedicaded to the Real Numbers. It introduces firstly integers and rational numbers, then it introduces the irrational numbers, which appeared in the first place as a necessity of expressing the length of incommesurable line segments. To my understanding, all of this helps to introduce the notion of continuity of a variable, which is put into work when treating the continuity of a variable dependent of another variable (if we restrict ourselves to single-valued functions).
The Archimedean Property is stated as follows (or as it is written in my textbook):
Given any number $c > 0$, there exists a natural number $n$, such that $n > c$.
Given any positive number $\epsilon$, there always exists a natural number $n$ such that the inequality $1/n < \epsilon$ is fulfilled.
The last statement is the first one with $c = 1/\epsilon$. Although I understand what is meant by it (I think it can readily be proved by contradiction), what I'm not getting is how the Archimedean Property fits in the preparatory material preceding that dealing with supremum, infimum and then with continuity and limits.
Does it have something to do with the concept of limit, perhaps?