Principle of mathematical induction

Principle of mathematical induction, sometimes referred to as strong mathematical induction, is as follows:

Let \(S\) be a subset of \(\mathbb{N}\) with these two properties: 1. \(1\in S\), and 2. for all \(n\in \mathbb{N}\), if \(n\in S\), then \(n+1\in S\) Then \(S = \mathbb{N}\).

In other words, to prove the statement: \(p(n)\) is true for all \(n\in \mathbb{N}\) - \(p(1)\) is true (base) - If \(p(j)\) is true for all integers \(1\le j \le m\), then \(p(m+1)\) is true