an example of mathematical induction

Below is a step-by-step demonstration of how mathematical induction (the Principle of Finite Induction) works.

PropositionPlanetmathPlanetmathPlanetmath. For any positive integer n, 2n-1n!.


Setup. Let S be the set of positive integers n satisfying the rule: 2n-1n!. We want to show that S is the set of all positive integers, which would prove our proposition.

Initial Step. For n=1, 2n-1=21-1=20=1, while n!=1!=1. So 2n-1=n! for n=1 and thus 2n-1n! all the more so. This shows that 1S.

InductionMathworldPlanetmath Step 1. Assume that for n=k, k a positive integer, 2n-1n!. In other words, we assume that kS, or that 2k-1k!.

Induction Step 2. Next, we want to show that k+1S. If we let n=k+1, then by the assumptionPlanetmathPlanetmath of the proposition, showing n=k+1S is the same as showing 2n-1n!, or 2k(k+1)!. This can be done by the following calculation:

2k = 2k-12 (1)
k!2 (2)
k!(k+1) (3)
= (k+1)!, (4)

where Equations (1) and (4) are just definitions of the power and the factorialMathworldPlanetmath of a number, respectively. Step (3) is the fact that 2k+1 for any positive integer k (which, incidentally, can be proved by mathematical induction as well). Step (2) follows from the induction step, the assumption that we made in the Induction Step 1. in the previous paragraph. Because 2k(k+1)!, we have thus shown that n=k+1S, proving the proposition. ∎

Remark. All these steps are essential in any proof by (mathematical) induction. However, in more advanced math articles and books, some or all of these steps are omitted with the assumption that the reader is already familiar with the concepts and the steps involved.

Title an example of mathematical induction
Canonical name AnExampleOfMathematicalInduction
Date of creation 2013-03-22 15:39:46
Last modified on 2013-03-22 15:39:46
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Example
Classification msc 00A05
Classification msc 00A35
Classification msc 03E25
Related topic PrincipleOfFiniteInduction