The first mathematical induction can be summarized as the following three steps:

(1) inductive basis: prove that the proposition holds when n= 1;

(2) Inductive hypothesis: the proposition holds when n=k is assumed;

(3) Inductive recursion: The proposition also holds when n=k+ 1 is deduced from inductive hypothesis.

It can be concluded that this proposition is applicable to all positive integers.

Proof of the correctness of mathematical induction;

Suppose we have completed the following reasoning.

Inductive basis: P(0) truth;

Inductive reasoning: For any k (P(k)→P(k+ 1))

But not all natural numbers have the property of p.

These natural numbers that do not satisfy the property p constitute a subset of non-empty natural numbers, so that there must be a minimum natural number in the subset, which is set as m.

Obviously m>0 is recorded as n+ 1, so n must have the property p, that is, P(n) is true.

Have n(P(n)∧? P (n+ 1)) for any k (? P(k)∨P(k+ 1)) is not satisfied ╞╡ not satisfied with any K (P(k)→P(k+ 1)).

The result of hypothetical reasoning contradicts the completed inductive reasoning, so the hypothesis is wrong.

All natural numbers have the property p.