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I think it's right
Edit:
TIL: when saying random numbers, some people think to integers, others to real numbers.
I also think that's correct... if we are talking about real numbers.
People are probably thinking about integers. I'm not sure about OP.
EDIT: I think it also works with p-adic numbers.
Yes real numbers, but as far as I'm aware it'll happen for integers too almost surely
I think you're confusing "arbitrarily large" with "infinitely large". See Wikipedia Arbitrarily large vs. (...) infinitely large
Furthermore, "arbitrarily large" also does not mean "infinitely large". For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers (as well as all other integers) are finite.
For integers I disagree (but I'm not a mathematician). The set of integers with infinite digits is the empty set, so AFAIK, it has probability 0.