The rigorous explanation for why 0.999…=1 is that 0.999… represents a geometric series of the form 9/10+9/10^2+… by definition, i.e. this is what that notation literally means. The sum of this series follows by taking the limit of the corresponding partial sums of this series (see here) which happens to evaluate to 1 in the particular case of 0.999… this step is by definition of a convergent infinite series.
You are just wrong.
The rigorous explanation for why 0.999…=1 is that 0.999… represents a geometric series of the form 9/10+9/10^2+… by definition, i.e. this is what that notation literally means. The sum of this series follows by taking the limit of the corresponding partial sums of this series (see here) which happens to evaluate to 1 in the particular case of 0.999… this step is by definition of a convergent infinite series.