To play the game FADEOUT, you begin by writing any set of 1994 numbers on the blackboard. You are then permitted to erase any pair of them, say a and b, and replace them by the number . You continue this process 1993 times, after which only one number remains on the blackboard.
Suppose that at the beginning of the game, all 1994 numbers on the board are 1. Prove that the number left at the end will never be less than 1/1994.