I do not agree with any of your statements. (1) There is no such thing as an *infinitesimal*. (2) 1-0.999... is not an infinitesimal, it is an indeterminate real value greater than 0. If this shocks you, why does it not also shock you that pi is also indeterminate. You can only approximate pi. (3) Therefore 1-0.999... > 0. 

Tango: There are infinitely many numbers between 0.999... and 1 - check the archives and you will find a clear example. You are incorrectly assuming that the limit of a series determines a number. If you insist on defining a number by a series (in fact this is circular), you must be consistent: what is the limit of any irrational number defined as a Cauchy sequence? You do not know, therefore you cannot use this definition. Furthermore, the apparatus for comparing numbers in decimal form must be consistent. You start wih the most significant digit in the representation and continue the comparison until you find two digits that are not equal. Having 0.999... = 1, contradicts this method. You must also remember that it is impossible to represent all numbers in any radix system. Most numbers can only be approximated. And if this is so, why would anyone foolishly insist on having 0.999... = 1? 98.195.24.26