The method of induction
From the concept of induction, the idea of mathematical induction appears, which allows demonstrations by the induction method
You can express the natural numbers as a triplet (\mathbb{N}_0,0,s) with \mathbb{N}_0 a set, 0\in \mathbb{N}_0, s| \mathbb{N}_0\rightarrow\mathbb{N}_0\backslash\{0\} an injective application, and in such a way that the fifth of the previous axioms is fulfilled, which is called the axiom of induction or the principle of induction
To do this, S is taken as the set of natural numbers that satisfy a certain property that wants to be proved, it is verified that zero fulfills the property (that is 0\in S), and that if a number n fulfills it, the next will also fulfill it (that is n\in S\Rightarrow s(n)\in S); and as a consequence of the axiom of induction, all natural numbers fulfill the property (S=\mathbb{N}_0)
In practice, the principle of induction is usually applied in terms of properties rather than in terms of sets. To carry it out, we will carry out the following steps:
Suppose that for each natural number n\geq n_0 it has a certain property P_n which may or may not be true. We assume that:
- P_n it is certain
- If for some n\geq n_0 the property P_n is true then P_{n+1} also what is
Then P_n it is certain for all n\geq n_0
The demonstration will be over, because we will have been able to test the property for all natives