Let's say you are betting on the outcome of some random variable. You can invest any amount $X$, and each outcome multiples that amount $X$ by a certain amount $ \ge 0$. You must choose $X$ less than how much capital you have. A Kelly Bet is one in which you bet an amount that maximizes the logarithm of your capital. The Kelly Bet will allow your capital to grow larger than any other betting scheme in the long term (i.e., as the number of bets approaches infinity).
What is the proof that previous statement?