>>13186414You don't even need the reals for this, considering all the quantities here are rational and dictated by geometric sequences. Sure if you have a hardon for needing to prove the geometric series converges you can use limits.
What really matters is that a multiple of the number you're finding the reciprocal of is congruent to -1 mod 10. Then you can just use long division to show that the number you used to obtain the multiple appears repeating in the expansion of the reciprocal. For example: . Do the long division yourself, I'm not formatting it in LaTeX.
You'll find that (and it's not difficult to prove that it's the case) for numbers which include primes other than 2 and 5 in their prime factorizations, you will get a repeating string. This repeating string is precisely the expansion of that rational number and contains no more "information" than a single repeating block and the fact that it repeats.
You can demonstrate the validity of arithmetic on these sequences by translating them with powers of 10 and subtracting off repeating portions. For example: . This reversible by using the process stated above.
Though I haven't rigorously proved it, it is at least convincing that . Either I am making no errors, or I'm making two errors that correct each other.
>>13186470Not even Burger believes this for repeating decimals.
https://youtu.be/YYnYatWV-pU?t=1521 
