Basic Properties of Initial Segments of the Natural Numbers

Let $\mathbb{N}$ be the set of natural numbers with successor map $S$, let $\le$ be the order on $\mathbb{N}$, and let $[n]$ denote the initial segment determined by $n$. Then the following hold for all $m,n,t\in\mathbb{N}$.

1. $1\in[n]$ and $n\in[n]$; in particular $[n]$ is nonempty.
2. $[1]=\{1\}$.
3. $S(n)\notin[n]$, and

4. If $m\le n$ then $[m]\subseteq[n]$.
5. The map $j\mapsto m+j$ is a bijection from $[t]$ onto $[m+t]\setminus[m]$. Moreover $[m]\subseteq[m+t]$, so $[m+t]$ is the union of the two disjoint sets $[m]$ and $[m+t]\setminus[m]$.