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induction - Proving Theorems Using the Principle of...

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Unformatted text preview: Proving Theorems Using the Principle of Mathematical Induction The Principle of Mathematical Induction deals with the problem of proving an infinite number of statements. Obviously it is impossible to prove each one separately, so we need a method of proving all of them simultaneously. The situation of interest here is when each statement is one which gives a fact involving a specific integer, and all of the statements are the same except for the integer involved. Furthermore, there is some first integer n for which we want to prove the statement, and then we want to prove it for all integers following n . We use the notation P ( n ) to designate the statement about integer n we wish to prove. Thus the ultimate goal is to prove all of the statements P n , P n 1 , P n 2 , …. There are many formats for induction proofs, and one can become less rigid in approach as one’s mathematical competence increases. The following format is suggested for those just learning mathematical induction. It includes all of the features which must be in any such suggested for those just learning mathematical induction....
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