My friend claims, that the digits in the decimal representation of pi contains every finite sequence of digits. For example my phone number will occur eventually. He claims that this is because there are infinite digits and they are non-cyclic.
While I agree that a decimal representation pi does have an infinite amount of non-cyclic digits, I am not convinced that these conditions are sufficient to imply that every finite sequence of digits are contained.
For instance, if we consider the decimal representation of pi: $$3.14159265358979...$$
If we (quite artificially) omit all occurrences of the digit $1$, we get: $$3.459265358979...$$
Or if you don't like that, we replace all occurrences of the digit $1$, with the digit $0$, we get: $$3.04059265358979...$$
I claim that (in both cases), we now have another sequence that is both infinite and non-cyclic. But clearly, the new decimal representation will not contain all finite sequences of digits. For example my phone number (which contains the digit $1$), will never occur.
I presume that the original claim is probably true, but the reasoning is (as demonstrated) apparently unsound.
My question is, is there some other condition of the decimal representation of pi, that is sufficient to guarantee that every finite sequence is contained?
Is this condition a property of all (natural) irrational numbers or transcendental numbers?
I don't think this related question is same, as it refers to an infinite random sequences.