Proof that the Kelly Criterion is optimal? 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?
 A: The statement is false.
For example consider the bet where you double your money with probability $2/3$ and lose it otherwise. Now consider the strategy "bet \$1 more than the Kelly bet on the first turn, and then use the Kelly bet after that". If this wins on the first turn it will forever have more money than the Kelly strategy, and if it loses on the first turn it will forever have less money than the Kelly strategy. So no matter how large the number of bets becomes, it has a $2/3$ chance of having a larger amount than the Kelly strategy.
A: The proof is in your link, Kelly's original paper and many subsequent papers but the key part here is the number of bets must very large. If you were bet-limited to, say, 4 bets to earn 16x your money, you're forced to bet your whole bankroll on double or nothing for any chance of success. That doesn't violate Kelly.
In fact, Thorpe of Fortune's Formula fame briefly explains on pages 11 and 12 that the optimal betting pattern for limited bets is not necessarily Kelly. Per your phrasing, Kelly is correct. Further, the capital growing larger than any essentially different betting pattern means Kelly will achieve a targeted capital size in the fewest bets.
This "essentially different" betting pattern refutes Oscar Cunningham's answer:
a) If one dollar is insignificant to your total bankroll / capital then it's same essential strategy.
b) If one dollar is, in fact, significant then that $1 extra is overbetting. Betting > 1.0 Kelly on even 1 bet reduces the growth rate of your capital while increasing risk of ruin. It is suboptimal in the long run.
Betting fractional Kelly is actually far more common than full Kelly, in order to reduce the risk of ruin. The growth rate is reduced less significantly in comparison.
A: Say in total I have 1 dollar to start with and I bet $X$ amount of my current net worth each time. Say if I win a bet (with probability $p$), I will get $X b$ back (net odd of $b$). Then after the bet with probability $p$, I got back
$1+Xb$
and with probability $1-b$, I will have
$1-X$ 
left. 
Now, consider betting for $N$ times, then there will be about $Np$ win and $N(1-p)$ loss. So, the final amount is about
$(1+Xb)^{Np} (1-X)^{N(1-p)}=((1+Xb)^{p} (1-X)^{1-p})^N.$
To maximize this gain, we just need to maximize $(1+Xb)^{p} (1-X)^{1-p}$ with respect to $X$. Or we can maximize $f(X)=\ln [(1+Xb)^{p} (1-X)^{1-p}]$ instead. Setting $\frac{d f}{d X}=0$, we have $\frac{pb}{1+Xb}-\frac{1-p}{1-X}=0\Rightarrow X= \frac{bp-(1-p)}{b}.$
