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Numerical Linear Algebra with Applications by William Ford

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Appendix B

Mathematical Induction

This appendix is a brief disussion of the topic, and is intended to be sufficient for the times in the book that a proof uses mathematical induction.

Suppose you are given a statement, S, that depends on a variable n; for instance,

1+22++n2=nn+12n+16,n1

si1_e

Let n0 be the first value of n for which S applies, and prove the statement true. This is called the basis step. For our example, n0 = 1. Now, assume S is true for any n ≥ n0 and prove that this implies S is true for n + 1. This is called the inductive step.

Then,

 S is true for n0, so S is true for n1 = n0 + 1.

 S is true for n1, so S is true for n2 =  ...

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