Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do not answer with the very common all horses are the same color example.]

Comment: Sometimes inductive arguments can lead to controversial conclusions, such as the surprise exam paradox, Richard’s paradox and a host of other paradoxes. However, I am interested in examples of a more mathematical nature (as opposed to linguistic) where the inductive argument is subtly flawed and leads to erroneous conclusions.

Note: If you provide an answer, please do so in a way similar to how current answers are displayed (gray out the flaw so people can be challenged to discover it).

Answer

Claim: ddxxn=0 for all n0.

Base case: (n=0): ddxx0=ddx1=0

Inductive step: Assume that ddxxk=0 for all kn. Then by the product rule,

ddxxn+1=ddx(xnx1)=xnddxx1+(ddxxn)x1=xn0+0x1=0.

Flaw:

In order for this to be a valid proof, the inductive step must be valid for all n0. However, when n=0, one can’t use the inductive hypothesis to rewrite ddxx1 as 0.

This “spoof” appears in Martin V. Day’s “An Introduction to Proofs and the Mathematical Vernacular.” Day gives its source as Edward J. Barbeau’s “Mathematical Fallacies, Flaws and Flimflam.”

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