Proposition: All natural numbers are interesting.

Proof by contradiction:
 

• Suppose the proposition is false.
• Then there exists a set S of uninteresting natural numbers.
• Since the set S consists of positive integers, the set is bounded below, and has a smallest element.
• The smallest element would constitute the smallest uninteresting natural number.
• Such an element, if it existed, would be very interesting indeed.
• By virtue of being interesting, such a number can not be uninteresting; the set S must be empty: a contradiction.

 QED
 

 Note: I think I heard this from Stan Ulam, but thanks to Bill Lindgren for pointing out that it appeared in Edwin Beckenbach: "Interesting Integers", Amer. Math. Monthly 52(1945), p211.