First, for any natural number n, let T(n) represent the nth smallest natural number and satisfy the property p. We want to prove that when n= 1, T(n) satisfies the property p. Because when n= 1, T( 1)= 1.
Because T(k) satisfies the property p, there is a natural number m, so T(k)=m and m satisfies the property p. What we want to prove is that when n=k+ 1, T(k+ 1) satisfies the property p. Because T(k+ 1) satisfies the property.
If t (k+ 1)
Knowledge expansion:
Mathematical induction is a mathematical method used to prove propositional identities or inequalities. It is based on the correctness of an initial proposition, which is proved to be true in all natural numbers by inductive reasoning. Specifically, mathematical induction includes two steps:
Initial step: prove that the proposition holds when n= 1 This is the starting point of mathematical induction.
Inductive steps: Assume that the proposition holds when n=k, and prove that the proposition also holds when n=k+ 1 This is the key step of mathematical induction. Using the hypothesis that the proposition holds when n=k, it is proved that the proposition holds when the next natural number n=k+ 1.
Mathematical induction is widely used to prove various propositions, such as divisibility, arithmetic progression's general formula, convergence of sequence and so on. The following is a simple example of proving arithmetic progression's general term formula by mathematical induction:
Example: prove arithmetic progression's general formula an=a 1+(n- 1)d by mathematical induction (where d is the tolerance).
(1) When n= 1, a 1=an, arithmetic progression's general formula holds.
(2) Suppose that when n=k, arithmetic progression's general formula holds, that is, AK = a1+(k-1) d.
Then when n=k+ 1, there is AK+1= AK+d = a1+(k-1) d+d = a1+(k+1)
To sum up, through the initial steps and inductive steps, we can prove that arithmetic progression's general term formula is valid in all natural numbers.
It should be noted that when using mathematical induction, we must ensure that the initial steps and induction steps are correct, otherwise the correctness of the conclusion will not be guaranteed. At the same time, mathematical induction needs to be used together with other mathematical methods, such as analysis and synthesis.