[Dov]

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Infinity plus or minus any finite sum is still just infinity. So an infinite bankroll cannot be increased (or decreased) with the martingale system if you are betting finite sums.

If you have a finite bankroll, using the martingale system is demonstrably -EV in any game like roulette where each trial is -EV.

So either way -- finite bankroll or infinite bankroll -- you cannot turn a -EV game into a +EV game by martingaling it.

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Why do you say that you can use a finite bankroll over an infinite time period?

Surely, if the time period is infinitely long, then you can't increase it by another day.

Similarly you can't have an infinite number of trials either, because you would never be able to increment the number of completed trials by 1.

Why do you accept an increasing number of trials, but not an increasing bankroll?