OK then. Since you agree 0.999... is equal to 1, let's use that to prove the other point:
Suppose (to make the maths easier) you're rolling a 10 sided dice, and you only lose if you roll a 6.
After the 1st roll, the probability that you won is 0.9 (there's a 1 in 10 chance that you rolled a 6)
After the 2nd roll, the probability that you won at least once is 0.99 (there's only a 1 in 100 chance that you rolled a 6 both times)
After the 9th roll, the probability that you won at least once is 0.999999999 (nine 9's)
If you play forever, the probability that you win at least once is 0.999... recurring forever.
You just agreed that 0.999... recurring forever is equal to 1, so the probability that you win at least once if you play forever is 1.
ie. you're going to win at least once, for sure.