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Let be the proposition that
is true, because
We assume that is true for an arbitrary integer with That is, we assume that for the positive integer with
We must show that under this hypothesis, is also true. That is, we must show that if for an arbitrary positive integer where then
We have for
This shows that is true when is true. This completes the inductive step of the proof.
We have completed the basis step and the inductive step. Hence, by mathematical induction is true for all integers with That is, we have proved that is true for all integers with