Principle of induction

 Let S be a subset of ℕ such that 

(i) 1∈ S, and 

(ii) if k ∈ S then k +1∈S. 

Then S = ℕ.

Proof.

 Let T=ℕ - S. We prove that T=Φ.

 Let T be non-empty. Then by the well ordering property of ℕ, the non-empty subset T has the least element, say m.

 Since 1 ∈  S and 1 is the least element of ℕ, m > 1. 

Hence m-1 is a natural number and m-1∉T. So m-1∈S.

 But by (ii) m-1∈ S ⇒(m-1) +1∈S, i.e. m∈S. 

This contradicts that m is the least element in T. Therefore our assumption is wrong and T=Φ

Therefore S= N. This completes the proof.