[chiralSPO]

0.9999.... is equal to exactly 1.

There are many ways of proving this using generally accepted and previously established methods, either algebraically or using limits. If you want to reinvent mathematics such that this is not the case, you may well find that your version of math gives paradoxical or nonsensical answers. (there are still some holes in our system, but overall it works pretty well, and this is not one of those holes--if you really want to tug at the very fiber of our system, look at the "axiom of choice" and the "continuum hypothesis.")

I would urge caution for those who have difficulty interpreting numbers that cannot be expressed perfectly decimal notation. Beware of irrational numbers (which cannot be expressed perfectly as a ratio of any two whole numbers), and especially of transcendental numbers (which cannot be expressed perfectly as a polynomial combination of rational numbers). These numbers are all very much real, easily defined (most of 'em), useful (some of 'em), and together vastly outnumber the "well-behaved" numbers...

[guest4091]

u =.1 + .01 + .001 +...+1/10^n

10u = 1 + u – 1/10^n

9u = 1  – 1/10^n

u = 1/9 – (1/10^n)/9

An 'infinite' sequence is never complete since 'infinity' is not a number.
Why did the early mathematicians define a 'limit' instead of declaring an equality?

[Bored chemist]

An 'infinite' sequence is never complete

Then 0.9999.... does not exist and we don't need to worry about it.
So, you can stop now.