I think it's darned near impossible, unless you are actually going to _use_ the definition in a proof of some non-obvious fact. A semi-obvious fact, such as the intermediate value theorem, won't quite do, for you now face the problem of explaining why this requires a proof at all. (Given a notion of continuity as “the function graph is connected”, it is not at all clear that this theorem is not obvious.)

Sorry to be such a downer – but you say it's a course on proofs, so maybe it's not so bad after all. I mean, it should be clear that the very idea of a proof requires definitions to work with? So you can tell them that if the need for this definition is not clear at present, it will become clear once you start trying to prove stuff. Mike's answer is of course excellent, if you can dig up the actual mistake Cauchy made and if it isn't going to be too advanced. All sorts of pathological examples, like space filling curves or functions that are everywhere continuous but nowhere differentiable, are good to drive home the point, but again there is a risk they might be too demanding.