Assume James the employee initially has $X$ dollars in his bank and each day he makes $S$ dollars in his job.
After each day of work he then wants to make a supplemental bonus income of $B$ dollars, to do this he bets $B$ dollars, and if he loses he bets another $2B$ dollars, and then keeps doubling until he wins, unless at some point he doesn't have enough money for the next bet, in which case he cries (all of these bets take place during the same day).
Please note when he bets $B$ dollars he gets $2B$ dollars if he wins and $0$ dollars if he loses (we assume both happen with probability $\frac{1}{2}$).
What is a super short way to prove James cries with probability $1$ regardless of the values of $X,S$ and $B$?
I am also looking for intuition as to why the strategy wouldn't work.
Here is my lame proof: The probability James doubles his money after reaching $K$ dollars is equal to the probability he has $D=\lceil\frac{K}{S+B}\rceil$ succesful days. In order for a day to be succesful he cannot lose more than $L=\lceil\log_2(\frac{K}{B}) +2\rceil$ times. Notice there is $C$ so that $D> 2^{L}/C$ for all $K$
Hence the probability of doubling is at most $(1-(\frac{1}{2})^L) ^ {2^{L}/C}$. This quantity is at most $e^{-(\frac{1}{2})^L(2^L/C)}= e^{-\frac{1}{C}}$. Therefore we double only a finite number of times with probability $1$.