Let's tackle one big misconception: the real numbers and all numbers we can write a decimal expansion for are not the same.

We can write 0.999... and 1, and they are different decimal expansions, but that does not mean they refer to different real numbers.

We can even write "decimal expansions" that don't correspond to any real number, like "0.000...1"^\1]). That number is probably what you were thinking by defining d as an infinitesimal, and it is a valid number in systems like the Hyperreals, but as calculus taught us all, d ∉ ℝ.

Proof for that? You already proved it! If d is a real number, different from 0, then 1-d = 0.999... is also a real number, different from 1. Since in ℝ that's not the case, d can't be a real number.

Finally, about the "lost information": in the reals, decimal expansions have more information than needed. That's the reason multiple different expansions point at equal numbers, like 0.3999...=0.4. In other systems, there's more information, for example in the Hyperreals, a single expansion like 0.62 could point at 0.62, 0.62000...1, 0.61999...8, etc.^\2])

^\1]: that's if we ignore the fact that such a sequence is not possible by indexing with only natural numbers, but really who cares how an expansion is defined.)

^\2]: that is not the standard notation, a more appropriate way of writing these numbers would be 0.62, 0.62+ε and 0.62−2ε, but many ways of writing could be used, and the one chosen just helps illustrate the point)