Recall pairing numbers: Given a pair of nonnegative integers, (m, n) there was a code number for the pair. The code number of (m, n) is denoted by m, n. The important part was that the function , is easily computable (we gave a simple formula, but the details do not matter at this point) and one-to-one and onto: Every pair has unique code number and every nonnegative integer is the code number of some unique pair. We can order the nonnegative integers lexicographically: m n if, and only if, m = m1, m2 and n = n1, n2 and (m1 < n1 or (m1 = n1 and m2 n2).

Now consider the following game of chicken: Player 1 chooses nonnegative integer p, q such that for left member p of the pair, p > 0, and puts a dollar in the pot. Player 2 must then choose another integer less than the previous choice, without restriction on the left member of the pair, in the lexicographic ordering and put a dollar in the pot. Play continues this way, without restriction on the left member of the pair, until one player chooses 0 (which is the least number in the lexicographic ordering according to our pairing number scheme). The player who first chooses 0 wins the money in the pot. There is a disincentive for choosing 0: more money accumulates in the pot each time a player doesnt choose 0. The interesting part is this: one player or the other must win.

(1) Play cannot continue forever.

Also, (2) there is no upper bound on how much can accumulate in the pot. Trillions of dollars could accumulate for example. Prove both (1) and (2).