I have a student who is working in their spare time on proving or disproving a conjecture of the form
$$\exists x.\forall y.\phi(x,y).$$
Right now their strategy is to construct an $x$ and then show $\forall y.\phi(x,y)$. So far so good.
The student has several times now brought me an $x$ and a "proof" that it works for all $y$ only for me to quickly point out a counterexample $y$. That's fine too; proof-writing can be tricky business.
But what's more worrisome is how they react to the counterexample: the $x$ the next time we meet will usually be a slight variation on the old $x$, adjusted just enough to handle the old $y$. Now since this problem is open, it's conceivable that such back and forth could eventually produce a valid proof. But I'm doubtful.
What I'd like to convey to the student is that they can't just react to the counterexamples I give them; they have to keep their eye on the bigger picture if they're going to inoculate their $x$ against any conceivable $y$. But so far these conversations have had little effect.
I could fairly easily just keep generating counterexamples and hope that they'll realize what's happening. But is there perhaps a nice exercise or analogy that would illustrate the issue faster and more clearly? Or should I focus on the trouble they seem to be having finding counterexamples for themselves?