For a number to be non-infinite, there must be at some point be a digit where all digits after it generate a 0.

For all numbers in our sequence, the probability of generating a 0 is 1/10: there is no point at which we cannot generate a 0. Furthermore, after the first 0 is generated at a, the odds of a+1 being 0 are also 1/10, as are the odds of a+2, a+3, and a+n. So we cannot identify a b, such that entry a+b must be >0, since the odds of any given a+b generating 0 are also 1/10.

the odds of randomly selecting 0 exactly an infinite number of times is exactly zero which is why OP is right

Probability of a=0 is (1/10)

• Pa = (1/10)
• Pb = (1/10)

Probability of both being 0:

• Pa AND Pb =(1/10)*(1/10)

then for n 0s

Pn = (1/10)^n

as n -> inf, Pn -> 0

put another way, (1/10)^inf = 0