Mathematics-Online lexicon: Mathematical Induction

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	Mathematics-Online lexicon:
	Mathematical Induction

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overview

Statements with natural numbers as their parameters can be proved by the Principle of Mathematical Induction. If is a statement that depends on $n\in\mathbb{N}$ , the method of proof consists of the following two steps:

Base step (also referred to as initial step): Verify that is true.
Conclusion (also referred to as inductive step): Show that the assumption ,, is true`` (induction hypothesis) implies the truth of , i.e.

$\displaystyle A(n) \implies A(n+1) \,.$

This establishes the truth of

for all $n\in\mathbb{N}$ .

The Principle of Mathematical Induction successively infers the truth of a statement from the previous statement . Therefore, if in the base step is verfied for some rather than , then the statement has only been proved for $n \geq n_0$ .

(Authors: Kimmerle/Abele)

see also:

Keyword: Numbers: natural numbers and integers

[Examples]

automatically generated 6/11/2007