The number won't be a real number. It might still be a more general kind of number.
Now there is the corresponding problem in surreal numbers. If we allow also transfinite integers as the integers in rational numbers of the form a/b does some "middle infinite" decimal expansion correspond to...
I am trying to prove that all numbers of the form 0.999... Squared end in a decimal value of 1. For example
0.99sq = 0.9801
0.999sq = 0.998001
Etc.
Is it possible to prove for all 0.999... ?
Let f(n) be of the form F(1) = 0.999999 F(2) = 0.9999999 F(3) = 0.99999999 Etc
For arbitrary N >= 1
Let g(n) = f(n)^2
G(1) and g(2) end in 8
By linear relation all g(n) end in 8
Assume 0.999... = 1
0.999... X 0.999... must end in 8 per (4)
1 x 1 does not end in 8
By (5),(6),(7)...