... work!''¹

Such an $n$ is called a counterexample.

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... case.²

Clearly, a simple induction is a special case of a strong induction.

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... induction.³

For complete examples of proof by induction, please see the Appendix.

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... shift.⁴

A more formal treatment would verify these properties of the inner while loop, probably by stating a separate loop invariant.

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... sorted.⁵

It might be better to terminate the recursion at, say, any array of size $< 15$ , because an alternate sorting algorithm, like Insertion Sort, will perform better for small $n$ .

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