It is a proof by contradiction: \(s-1<s\) prevents \(s-1\) from being an upper bound, and from \(n>s-1\) we get \(n+1>s\) with \(n+1\in\mathbb N\). Saying “it is infinite” or using the limit of \(1/n\) would assume what is to be proved.
The property of \(\mathbb R\) by which, however small a positive number \(x\) is, some multiple of it by a natural \(n\in\mathbb N\) exceeds any real \(y\).