Peano's Axioms

Following are the 5 Axioms of Peano or also called as Peano's Postulates
N1. $1 \in \mathbb{N}$
N2. If $n \in \mathbb{N} \Rightarrow n+1 \in \mathbb{N}$
N3. $1$ is not the successor of any element in $\mathbb{N}$.
N4. If two numbers $m,n \in \mathbb{N}$ have the same successor then $m=n$.
N5. A subset of $\mathbb{N}$ which contains $1$, and which contains $n+1$ whenever it contains $n$, must equal $\mathbb{N}$

 Q. What is the significance of Peano's Axiom ?
Most familiar properties of $\mathbb{N}$ can be proved using Peano's Axioms.

 Q. How do you prove N5 ?
Given the set contains $1$. If We will prove by Contradiction Suppose there is a set $S \subseteq \mathbb{N}$ and $S \ne \mathbb{N}$, that means there is a smallest element $n_0 \in \{n\in \mathbb{N}| n \not \in S \}$. Obviously $n_0 \ne 1$ as $1 \in S$. As $n_0$ is the smallest element which is not in $S \Rightarrow n_0-1 \in S$. But if $n_0-1 \in S \Rightarrow n_0-1+1=n_0 \in S$ so we have a $\Rightarrow \Leftarrow$ and our assumption that there exists a number outside set $latex S$ is False and $S=\mathbb{N}$