Nearly every classical or recreational math result about a finite set is an example.
Can we have a walk in Königsberg so as to cross each of its seven bridges once? Well, we can attempt starting at southeast point then cross northward then cross back then go east... Never mind, Euler had a better approach.
Can we tile a chessboard with A1 and H8 removed by $...
Prove that all popes had less than 150 years when they died.
You could look at each of 260 of them and verify that this holds.
Or you could take the fact that the oldest known person lived up to 122 years, which can be used to automatically conclude that all popes can only be less than this.