CONO-SUR 2011 Problem 2

The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$ . Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$ , no matter what numbers Pedro chose in each step.