When trying to prove a given statement for a set of natural numbers the first step known as the base case is to prove the given statement for the first natural number. The induction axiom in Peano Arithmetic says that for any predicate statement about numbers ϕ if you can prove ϕ 0 is true and you can also prove that for any number n ϕ n ϕ n 1 then ϕ n is true for all n. It has only 2 steps.
Show that given any positive integer n n n3 2n n 3 2 n yields an answer divisible by 3 3.
The most common form of proof by mathematical induction requires proving in the inductive step that k P k P k 1 displaystyle forall kPkto Pk1 whereupon the induction principle automates n applications of this step in getting from P 0 to P n. The principle of mathematical induction has a very special place in mathematics because of its simplicity and vast amount of applications. Then all are true. How to prove using the Principle of Mathematical Induction.