You could argue a lot about how [math]0.99 \dots[/math] is unclear, how can the dots be very ambiguous and how can one abuse notation. Well, maybe what is needed is to justify and explain notation and representation. To really understand numbers real analysis is needed, so you will lose your time with someone who only knows up to basic algebra because without any fundaments you can talk very few things about numbers and infinity.

Now, what is [math]0.9999999 \dots [/math] ?

[eqn]0. \bar{9} = \frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{9}{1000}+\frac{9}{10000} \dots[/eqn]

That is your answer. If you really want to know what the dots notation mean, how to take "infinity" into account, you would want to know calculus first and deal with the infinite series. But without topology and analysis, even without a hint of set theory and its axiomatizations, how would you say anything of what mathematicians understand of infinity, only with your "intuition" and your misleading use of notation. What is [math]0.999 \dots[/math] and does it is equal to [math]1[/math]? Solve this:

[eqn]0. \bar{9} = \sum_{k=1}^{ \infty} \frac{9}{10^k}=x[/eqn]

Where [math]x[/math] is your proved solution. But if you don't know what a convergences test or what is a geometric series is, you will only argue over your intuitive notion of the symbols and you will not be arguing over math at all. Semiotics is a thing but is not the central thing in math.

Nothing in the world, in science, in humanities and even in abstract systems like math should be like you want it to be, in a way that you feel good about. This desire is irrational. You must accept a fair scientific fact, even if you would want things to be different. You should distinguish between what you see and what you want to see and it is not very different if instead of science you are thinking maths.

"All the truths of mathematics are linked to each other, and all means of discovering them are equally admissible" Adrien-Marie Legendre.