Instructor: In fact this expression is equal to that expression. Let us see how we can convince ourselves that that is true.
Student: I'm more than willing to take your word for it! You're the expert!
This student feels that he is reassuring the instructor that his word will not be doubted, so he is sparing the instructor some work, reassuring the instructor that the instructor's goal is already accomplished, and he is sparing himself what he expects would be the agony of learning whatever it was the instructor was leading into.
How do we know that $\pi$ is irrational?
Well, duh!!! That's really basic. We learned that in 7th grade!! Everybody knows that!! Every textbook and teacher says so.
In fact, typical students don't know that the subject matter is precisely: how can we learn how to tell, independently of professors and authorities who tell us, what is true?
In order to explain that point to them, I would like some good examples, perhaps in elementary arithmetic, of situations where students as naive as the one quoted above (who is a fictitious composite of a number very real students) of easily explained problems in which even students like the one above would understand that the way they know the answer is correct is that they can see why it must be correct. That would illustrate the point that learning how to do that in more advanced subjects is precisely the subject matter of courses in those more advanced subjects.
Are there any sexy, or at least good, examples of that sort of problem, that could serve that purpose well?