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.
To apply the concept of the Intermediate Value Theorem to your situation, you can consider the following "existence proof":
1. Define a continuous function: Consider a function that represents the possibility of your sweetheart being home as a function of time. Let's say this function is continuous, which means that there are no abrupt changes or gaps in between.
2. Establish endpoints: Identify two time points, one representing the earliest possible time of arrival and the other representing the latest possible time of arrival.
3. Evaluate the function at the endpoints: Determine whether your sweetheart is home at the earliest and latest possible times of arrival. If your sweetheart is home at the earliest time but not at the latest time, or vice versa, it gives an "existence proof" that at some point in between the two times, your sweetheart must have either arrived home or left.
By applying the Intermediate Value Theorem, you can conclude that there exists a time interval between the earliest and latest possible times where you will find your sweetheart at home. This proof gives you confidence that it is worthwhile to make the trip.
However, this existence proof does not provide you with the precise time to visit or any information about factors other than your sweetheart's presence. For example, it does not consider whether your sweetheart might have prior commitments, unexpected plans, or any other constraints that could affect their availability.
Mathematicians are interested in conducting existence proofs before investing time in searching for solutions because it allows them to determine if a solution exists at all, without needing to find the exact solution. This approach helps save time and resources. By proving the existence of a solution, mathematicians can then focus their efforts on finding the actual solution, if necessary, through further analysis, calculations, or techniques specific to the problem at hand.