[math-fun] Request for example elementary non-constructive proofs with "witnesses"

Could anyone supply me with elementary examples that illustrate the idea of a non-constructive proof, for those with a "Martin Gardner reader" level of mathematical sophistication that also has a not-too-trivial but reasonably easily-verified case? For example a non-constructive proof that some set of numbers is non-empty, along with an example of such a number that can be fairly easily checked? Thanks!