An existence proof is simply a proof that shows a particular kind of object exists, or can exist. Why is that interesting? Because if you're in the early days of gathering data about something and you never observe the thing actually happening, it's very hard to tell if it's a 0 probability event (it will never, ever happen) or just a low probability event (it will happen, just very rarely). Once you have an existence proof you know you're not completely wasting your time by searching.
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