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).
Best Answer
Claim: $\frac{d}{dx}x^n=0$ for all $n\ge0$.
Base case: ($n=0$): $\frac{d}{dx}x^0=\frac{d}{dx}1=0$
Inductive step: Assume that $\frac{d}{dx}x^k=0$ for all $k\le n$. Then by the product rule,
$$\frac{d}{dx}x^{n+1}=\frac{d}{dx}(x^n\cdot x^1)=x^n\frac{d}{dx}x^1+\left(\frac{d}{dx}x^n\right)x^1=x^n\cdot0+0\cdot x^1=0.$$
Flaw:
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.”