I went back and edited my
proof. It's much clearer now, I think. I had been up for about 36 hours when I first wrote it. I think it's the clearest proof of it on this thread because it's a 'why' proof that actually explains why it is the way it is as opposed to relying on calculus like the OP's proof, a formula like Kromoh's proof or algebraic trickery like xei's proof. xei's proof is the most elegant though without a doubt. My proof is essentially along the same lines as xei's informal argument that almost has you convinced. Making it formal might help. Unfortunately, it's the longest proof on the thread, but that's almost always the way it is with explanatory proofs vs. clever proofs.