Since putting an infinite number of 9s onto the end of A_{n} cannot decrease its value, 0.999... is greater than every number that is less than 1.

Actually, if I'm reading it correctly you've shown that 0.999... is greater than or equal to every number less than 1. You are assuming that you're working in a group which is topologically continuous, but not showing it.

So suppose we defined a real number as a sequence (n,d1,d2,d3,...) where n is an integer and d1,d2,d3,... are integers such that 0<=d

_{i}<=9. You can work out operations to define a+b, a*b, etc. and the zero of the group would be (0,0,0,...)

Define (a,n1,n2,n3,...)=(b,m1,m2,m3,...) iff a=b, n1=m1, n2=m2, etc.

In that case we have (0,9,9,9,...) and (1,0,0,0,...) as distinct elements.