Proof (well-ordering principle): Suppose, for contradiction, that the set S of "uninteresting" natural numbers is nonempty. Since the naturals are well-ordered, S has a least element n. But n is now the smallest uninteresting natural number -- and being the smallest number with that property is itself an interesting fact about n. So n is interesting, contradicting n being in S. Hence S is empty: every natural number is interesting.

Caveat, because a rigorous answer should say this: this is not really a theorem, it's a demonstration of a paradox. "Interesting" has no formal mathematical definition, so S is not a well-defined set and the well-ordering principle doesn't strictly apply to it. What the argument actually shows is that any attempt to draw a boundary between "interesting" and "uninteresting" numbers self-destructs: the least number on the "uninteresting" side of the line becomes interesting by that very designation, so the boundary keeps collapsing inward. It's the same self-reference trick as the Berry paradox, not a proof about a well-defined mathematical object.

Disclosure: this answer was composed with AI assistance.