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Proof by induction is a proof made by proving that a statement is true for a general case (i.e. when EQN:n=k ), the next case and the first case; if all three of these are true, then, by induction, the statement must be true.

Proof by induction is a proof made by first assuming that a statement is true for a general case (i.e. when EQN:n=k ), then proving that it still holds true for the next case (i.e. when EQN:n=k+1 ) and then proving that it is true for the first (base) case; if the statement /does/ hold for /both/ the base case /and/ the inductive step ( EQN:n=k+1 ), then, by induction, the statement must be true.

!! METHOD

To prove that something is true for all integers EQN:n{\ge}r , first prove that it is true for EQN:n=k , then prove that it is true for EQN:n=k+1 , and finally prove that it is true for EQN:n=r.

To prove that something is true for all integers EQN:n{\ge}r :

* assume that it is true for EQN:n=k

* prove that it remains true for EQN:n=k+1

* prove that it is true for EQN:n=r.

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#Hello# #Josh#

case n = k is not proved but assumed.

The proof consists of two steps:

* The basis (base case):

** Show that the statement holds when n = 0 (or n = 1 ...)

* The inductive step:

** show that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.

Hope this helps

!! Examples

It would be nice to have some small, clean examples here. Not too many, not too much.