[jason1990]

Let p be the probability of winning the game and assume p<1. Let P_k be the probability that we win every game after the k-th game. If we can show that P_k=0 for all k, then the result is proven.

Let k be an arbitrary integer and let x be an arbitrary positive real number. Since p<1, the limit as n goes to infinity of p^n is 0. Therefore, there exists and integer n such that p^n<x. Let P_{k,n} be the probability that we win all the games from game number k+1 to game number k+n. Then

P_k < P_{k,n} = p^n < x.

So P_k<x for all positive numbers x. Hence, P_k=0. Since k was arbitrary, P_k=0 for all k, and this completes the proof.