Suppose we take N i.i.d. normal draws and look at the difference between the highest and the second-highest draw. Does this converge to a nonzero constant as N grows?

Answer: No.

Proof: if it did, we would all know this constant.

Got this as a puzzle by someone who thought "Yes." I was very surprised that my proof could be false but willing to accept it. A few days later, got an email saying, "okay, it decays to 0 but very, very slowly."

(Thanks @franktron for correcting an error!)

I'm learning that economists are quite interested in the difference between the highest and the second-highest thing...!