Silly proofs 1

My general style of proof in mathematics is to apply deep and powerful theorems in clever ways. I don't like going back to basics, and thus will often apply a big theorem in lieu of making explicit estimates, direct calculations, etc. Sometimes this gives you a better proof, other times it just gives you a shorter and more mysterious proof.

This, combined with my general liking of severe overkill, is probably what leads to me enjoying proving things in silly ways. I'm going to use this blog to share some of my favourite silly proofs.

This one came up recently:

Let M_n be a sequence of real numbers with M_n -> inf. Show that there exists a_n > 0 such that sum a_n converges and sum M_n a_n diverges.

Now, if you want to be boring you can construct such a_n explicitly. But my way is much more fun. "Ah ha!" I thought, "If it didn't then you could define a discontinuous linear functional on l_1, and we know that's impossible!". What follows is basically just a sketch of the most elementary version of this argument I could use:

Suppose not. Then for every a_n in l_1 we have sum M_n a_n converges (as it converges absolutely). Define f : l_1 -> R by f(x) = sum M_n x_n. Then f is linear and discontinuous - because ||f|| >= M_n -> inf.

Now, let f_N = sum_{n <= N} M_n x_n. f_N is continuous, and for all x we have f_N(x) -> f(x). So f is a pointwise limit of continuous linear functionals. But the continuous linear functionals are closed under pointwise limits (e.g. because they are Borel measurable, because of the uniform boundedness theorem, because the stars are in the correct alignment, etc), contradicting the fact that f is discontinuous!

I leave the elementary proof as an exercise for the interested reader.