Hmmm... I don't think your example proved that .999... was not equal to 1.
Assume "<>" means "does not equal"
0.1111... <> 1.1111...
One good thing about math is that it abstracts things. We abstract things to what we think - or would like to think - are the elements. We have reasonably well-defined symbols, a collection of symbol "sentences" that are true (or at least assumed true) and we have production rules for combining these sentences to produce new sentences that are also true. These symbols, sentences, and rules form what is known as a grammar.
It's easiest to use this grammar if one doesn't try to ascribe a physical or real interpretation to each of the productions, but only to the final state. (George Boole explains this in some detail - repeatedly - in his book "The Laws of Thought.")
Doing this invariably leads us to some startling conclusions. Despite this counter-intuitiveness, we learn to accept that some things are just not what they seem.
We trust the simplest rules applied to the simplest rules as opposed to the complicated thoughts - not always, but generally.
