Ones that are meaningful. These are not. As I said before, you don’t seem to understand how proofs are constructed or how they work. This constant idea that the digits have to be different is something of your own invention. Nice idea, and works a lot of the time. But not always true. Things just go around in circles.
0.99999… is notation for 9x10[sup]-1[/sup] + 9x10[sup]-2[/sup] + 9x10[sup]-3[/sup] + 9x10[sup]-4[/sup] + 9x10[sup]-5[/sup] + …
This is by definition. If you want to prove something, try sticking to this definition, and stop making up ideas about what is required for equality that are not part of well understood mathematics.
Equality over the real numbers is well defined. And it does not involve expressing a number a decimal and comparing digits. It involves showing that the difference between the two numbers is zero.
It appears that a lot of the issues with 0.9999… are coming from an unstated, and probably dimly realised point of view that the only numbers that “exist” are the rationals. This gets us back to the ancient Greeks. They were really upset when √2 was discovered. For much the same reasons I suspect. But almost all real numbers are irrational. That isn’t something that can be debated. Building the real numbers so that it can accept all these irrational numbers has, amongst other things, the result that there are no infinitesimals with a value other than zero. And 0.999… then equals 1.
As to mathematics as an experimental science - well as expressed many times above - it isn’t. It is much more rigorous. The way I like to explain the rigour and the power is this.
We can define the integers with just five axioms - the Peano axioms. Once we have them, we can define a few useful operations that make these numbers interesting and useful. Addition, subtraction, multiplication and division. Very quickly we get a few interesting results. The integers are not closed under division. 1/2 isn’t an integer. But we can also prove a few things. We can define the idea of odd and even numbers, and show that the sum of two odd numbers is never odd. This isn’t something that is subject to experiment. You can’t go out and keep looking for a pair of odd numbers that do sum to another odd number. We know you will never find them.
Similarly we can note that some numbers are equal to a group of other numbers all multiplied together. So we can define the notion of factors, and factorisation over the integers. All still based on only the Peano five axioms. Similarly we can note that some numbers are not so factorisable, and we can define the notion of prime and composite numbers. So, is there a limit to the number of prime numbers? Again, this isn’t something done by experiment. There is no point searching for the largest one. We can prove, from a basis of just those five axioms that there is no largest prime, and indeed they go on forever.
And my favourite. We can note that any integer can be expressed as a product of a group of prime numbers. So, is there more than one way to pick a group of primes to multiply together to obtain an integer? This is the unique prime factorisation theorem. And the answer is that no, you cannot find more than one way of factorising any integer into a group of primes (ignoring trivial reordering.) You can prove this. There is no point going out to try to find an integer that has more than one prime factorisation, no matter how hard or long you look, you will not find one. Absolute. No wiggle room. There is no point even trying once.