let 0<x show that there is a unique m in N such that m-1 < or = x < m.
the book says to consider n in N : x < n and that N is well ordered
For $\displaystyle x\in\mathbb{R}$, consider the set $\displaystyle S_x=\{n\in\mathbb{N}:n>x\}$. By the well ordering principle, this set has a unique smallest element. Call this element $\displaystyle m$.
$\displaystyle x<m$ because $\displaystyle m\in S_x$. $\displaystyle m-1\leq x$ because if it isn't, then $\displaystyle m-1\in S_x$, contradicting the fact that $\displaystyle m=\min(S_x)$.