posted by Desperate .
This is all about the intermediate value theorme..
You wish to visit you sweetheart, but you don't want to go all the way over to their house if your sweetheart isn't home. What sort of "existence proof" could you do beforehand to decide whether it is worthwhile to make the trip? What sort of information will your proof not give you about making the trip? Why do you suppose mathematicians are so interested in doing existence proofs before they spend a lot of time searching for solutions?
It's difficult to see how the problem as stated is related to the Intermediate Value Theorem - but putting that aside, presumably what you're looking for is a good indication that the sweetheart is at home without going over there. You could phone, of course, but if you wanted to be more circumspect about it you could consider (a) logging in to MS Messenger to see if he/she's online while appearing offline yourself, (b) phoning a mutual friend to see if they know the sweetheart's whereabouts, or (c) checking the TV guide to see if one of his/her favorite shows is on. Obviously none of these is conclusive, and none of them will tell you whether he/she actually wants to see you anyway, but answers to any or all of them might affect your assessment of the likelihood that making the trip will result in happiness all round - which is pretty much the same reason why mathematicians would be interested in knowing whether proving something is likely to possible before spending a lot of time and effort actually trying to do it.