[constantcrying]

>I'd say the key point is to understand the difference between a number, and the decimal representation

But there is no such distinction. In fact the decimal representation is "closer" to a real number, then just 1.

>is one possible representation of number 1.

Why? You are just asserting things. You do not even give an argument why that should be the case. Why is 0.999... a representation of 1 and not 0.123?

[yodsanklai]

> But there is no such distinction

Of course there's a distinction. A decimal representation is a sequence of digits, not a number

> Why?

It boils down from the definition of the decimal representation and the limit of a geometrical sequence.

https://en.wikipedia.org/wiki/Decimal_representation

[constantcrying]

>Of course there's a distinction. A decimal representation is a sequence of digits, not a number

Oh and what is a real number? Might a real number be a sequence of rationals? Or more correctly an equivalence class of cauchy series.

>It boils down from the definition of the decimal representation and the limit of a geometrical sequence.

No, it doesn't. You are assuming the conclusion.