# 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).

Claim: $$ddxxn=0\frac{d}{dx}x^n=0$$ for all $$n≥0n\ge0$$.

Base case: ($$n=0n=0$$): $$ddxx0=ddx1=0\frac{d}{dx}x^0=\frac{d}{dx}1=0$$

Inductive step: Assume that $$ddxxk=0\frac{d}{dx}x^k=0$$ for all $$k≤nk\le n$$. Then by the product rule,

$$ddxxn+1=ddx(xn⋅x1)=xnddxx1+(ddxxn)x1=xn⋅0+0⋅x1=0.\frac{d}{dx}x^{n+1}=\frac{d}{dx}(x^n\cdot x^1)=x^n\frac{d}{dx}x^1+\left(\frac{d}{dx}x^n\right)x^1=x^n\cdot0+0\cdot x^1=0.$$

Flaw:

In order for this to be a valid proof, the inductive step must be valid for all $$n≥0n\ge0$$. However, when $$n=0n=0$$, one can’t use the inductive hypothesis to rewrite $$ddxx1\frac{d}{dx}x^1$$ as $$00$$.

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.”