I'm currently the teaching assistant for a course that serves as an introduction to rigorous proofs, and I've noticed some of my students have a tendency to try and use a sort of "backwards reasoning" on induction problems. Specifically, if they were asked to prove something like $n! > n^2$ for $n \geq 4$, for the inductive step they will start by writing the very inequality they want to prove:
$$ (n+1)! > (n+1)^2.$$

It's *not* just a case of assuming the conclusion however, because they'll then manipulate this inequality until they reach some new inequality they know is true, a la
\begin{align}
(n+1)n! &> (n+1)(n+1)\\
n! &> (n+1)
\end{align}
And then they'd say something like $n! > n^2 > n+1$, with this last inequality being something they'd already proved.

The problem then is that it's clear from looking at their work that many of them are getting tripped up by this method. Further, I doubt that most of them realize that this only works because each step in the chain of reasoning is *reversible* (at the very least, none of them made it explicit that they were relying on this fact).

I'd like to convince them that if they want to use this method, they need to be very careful. It would help emphasize my point if I could present an induction problem that would be difficult to answer using this method. In particular, I've been trying to think of a way to force them to use a step that is not reversible, so that this backwards reasoning leads to stating something that does not follow from their previous statement. Currently I'm messing around with things like $z^2 > 0$ followed by $z > 0$, but haven't found a problem that I feel is adequate yet. Any thoughts?