Suppose a game where a fair coin is flipped. Tails pays out $0.5$x the bet, and heads pays out $1.75$x. With a bet of \$1, the expected value of a single game is $0.5\cdot 0.5 + 0.5\cdot 1.75 = 1.125$. So if $1000$ games of this is played, \$125 is expected to be gained.
But what if instead of betting \$1, the entire bankroll is bet? So starting with \$100, after a single round there is a $50$% chance of \$50, and a $50$% chance of \$175. Regardless of outcome, the entire amount is bet again, so if it was heads, the next round \$175 would be bet and so on. What is the expected amount after $1000$ rounds?
My intuition is that you'd end up with \$0, since it would take more than one win to make up for a single loss as $0.5\cdot 1.75 = 0.875$, and when I run a few trials programmatically it would come out to $0$ (as in, $<\epsilon$) each time, but I don't know how to prove this