[mr_mitm]

Any confusion about this should go away as soon as you make clear what exactly you are talking about. If you construct the real numbers using Cauchy sequences and define the* decimal representation of a number using a Maclaurin series at x=1/10 then it's perfectly clear that 0.9... and 1.0... are two different representations of the same number. So it's the same equivalence class, but not the same representation. Thus, if you're talking about the representation of the abstract number 1, they're not equal but equivalent. If you're talking about the numbers they represent, they're equal.

* As the example shows, the decimal representation isn't unique, so perhaps we should say "_a_ decimal representation".

[dagw]

The intersection between people who are both confused by this and are comfortable working with Cauchy sequences, Maclaurin series and equivalence classes, is probably pretty small.