2 comes after 1 in the natural numbers. 2 comes after 1 in the set of integers. What number comes after 1 in the set of reals? Its not 2, because 1.5 exists. Its not 1.1 either. Nor is it 1.000000001. The number that comes immediately after 1 in the real numbers can then only be 1.000...01, the number which is infinitesimally larger than 1. This infinitesimal value is 0.000...1, since 0.000...1 + 1 = 1.000...1

Consider the number that comes before 1 now. This number should be ininitesimally smaller than 1. 1-0.000...01 = 0.999...9. Since this number comes before 1, it must not be the same as 1. Thus 1=/=0.999... QED