>>14324673You are still thinking of the wrong process. The process is not
0.9
0.99
0.999
0.9999
...
Instead it is
0.999... is a value between 0.000000... and 0.999999... inclusive
0.999... is a value between 0.900000... and 0.999999... inclusive
0.999... is a value between 0.990000... and 0.999999... inclusive
0.999... is a value between 0.999000... and 0.999999... inclusive
and so on.
This is vacuous until we decide that 0.000... = 0 and 0.999... = 1, at which point we have the definition
0.999... is a value between 0 and 1 inclusive
0.999... is a value between 0.9 and 1.0 inclusive
0.999... is a value between 0.99 and 1.00 inclusive
0.999... is a value between 0.999 and 1.000 inclusive
and so on.
We can compare this to the process of "knowing the answer was exactly 1 from the beginning":
1 is a value between 1 and 1 inclusive
1 is a value between 1 and 1 inclusive
1 is a value between 1 and 1 inclusive
...
Notice that at each stage the intervals generated by the 0.999... and 1 processes have a common value, 1. Therefore by the definition in
>>14324582 they are equal.
