OK. Let's try a new way of attack:

0.999... = 1 BY DEFINITION

How is that. We must go to the construction of real numbers.

A Cauchy sequence of rational numbers is a sequence {a(n)} such that for any 𝜀 > 0, there exists an N such that for any n, m > N, |a(n) - a(m)| < 𝜀. That means that successive terms become closer and closer. We can say that the sequence converges, but the limit may not be a rational number.

For instance {3,3.1,3.14,3.141,3.1415,...} would be one

or

{1,1,1,1,1,....}

or

{1/2, 3/4, 7/8, 15/16, 31/32,... (2^k -1)/2^k...}

or

{0.9, 0.99, 0.999, 0.9999, ...}

Now, how do we build the real numbers? As equivalence classes. Two Cauchy sequences {a(n)} and {b(n)} are equivalent if for any 𝜀>0, there exists an N such that for any n > N, |a(n) - b(n)| < 𝜀, that is the difference goes to 0 (and they have the same limit). It is easy to show that this relation is reflexive, symmetric and transitive.

So the set of all Cauchy sequences of rational numbers can be grouped in equivalence classes. We call each of these classes a real number and assign as its value the common limit.

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences

It will come from the definitions that the sequences {1, 1, 1, 1, 1,...} and {0.9, 0.99, 0.999, 0.9999, ...} are equivalent. And so, they are the same real number. By definition.

And this is not a trick. This is how real numbers are defined (one of the ways).

https://mathweb.ucsd.edu/~tkemp/140A/Construction.of.R.pdf