Or, at least, that’s what this blog post at Science and Math Defeated aims to do. Normally, I avoid writing on such a topic but I think the following example could be instructive to a few people, at least, in learning how not to infer from mathematical induction. The author of that blog post sets to “disprove” the foundation of Calculus by showing that the “assumption” $0.999 \ldots = 1$ leads to a contradiction (which I am sure most of you have seen before.) And this is supposed to be achieved  through the use of Mathematical Induction.

Let $P(n)$ be the statement $\displaystyle 0.\underbrace{999...9}_{n \, 9's} < 1$ for all $n \in \mathbb{N}$ and $n \ge 1$.

Claim: $P(n)$ is true for all $n \ge 1$.

Proof: $0.9 < 1$, and so, $P(1)$ is true. This takes care of the base case. Now assume $P(n)$ is true for some $k \in \mathbb{N}$, where $k \geq 1$. Now, it is easy to show that $P(k+1)$ is true as well (I just skipped some details!). Hence, $P(k) \Rightarrow P(k+1)$ holds. This takes care of the induction step. (Note that $P(k+1)$ is shown to be true independent of $P(k)$!) And, this proves our claim.

(Erroneous) Conclusion: Hence, $0.999\ldots < 1$.

Notwithstanding the inductive proof (which is correct) above, why is the above conclusion wrong?

Ans. Because “infinity” is not a member of $\mathbb{N}$.

(Watch out for Todd’s next post in the ETCS series!)