Originally Posted by
tomcowley
You're trying way too hard- the general point doesn't involve "almost surely". If you start with $100, and play an infinite series of coinflips, losing costs you $1, winning wins you $1000, what are the odds of ever dropping below $100 on that infinite sequence? The odds, obviously, are "just over 50%", and not anywhere near 100%. If you lose the first flip (50%), you drop below, but if you win the first flip, then you're a gigantic underdog to ever drop back to $100.