Mathematical induction

This is an extremely powerful technique of mathematical proof, used at all levels.

You want to prove a proposition which depends on an integer n. First you prove the proposition for a particular value of n, say n=0 or n=1.

Then you prove that if the proposition is true for a value n, then it must be true for n+1. Those two steps together mean that you have proved the proposition for all n from 1 upwards.

Formally, you must formulate the thing that you want to prove in the form of a proposition P_n that depends on n. There are then two steps:

• You must prove P_a is true for some a
• You must prove P_n ⇒ P_n+1

You have then proved that P_n is true for all n≥a

Induction is a very powerful tool for proving things, although it doesn't usually help you to find the propositions that you are trying to prove.

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