It’s almost sure to be the case, but nobody has managed to prove it yet.
Simply being infinite and non-repeating doesn’t guarantee that all finite sequences will appear. For example, you could have an infinite non-repeating number that doesn’t have any 9s in it. But, as far as numbers go, exceptions like that are very rare, and in almost all (infinite, non-repeating) numbers you’ll have all finite sequences appearing.
Yes, compared to the infinitely more non exceptions. For each infinite number that doesn’t contain the digit 9 you have an infinite amount of numbers that can be mapped to that by removing all the 9s. For example 3.99345 and 3.34999995 both map to 3.345. In the other direction it doesn’t work that way.
It’s almost sure to be the case, but nobody has managed to prove it yet.
Simply being infinite and non-repeating doesn’t guarantee that all finite sequences will appear. For example, you could have an infinite non-repeating number that doesn’t have any 9s in it. But, as far as numbers go, exceptions like that are very rare, and in almost all (infinite, non-repeating) numbers you’ll have all finite sequences appearing.
Exceptions are infinite. Is that rare?
Yes, compared to the infinitely more non exceptions. For each infinite number that doesn’t contain the digit 9 you have an infinite amount of numbers that can be mapped to that by removing all the 9s. For example 3.99345 and 3.34999995 both map to 3.345. In the other direction it doesn’t work that way.
Very rare in the sense that they have a probability of 0.