Unfortunately not an ideal proof.

It makes certain assumptions:

1. That a number 0.999… exists and is well-defined
2. That multiplication and subtraction for this number work as expected

Similarly, I could prove that the number which consists of infinite 9’s to the left of the decimal separator is equal to -1:

...999.0 = x
...990.0 = 10x

Calculate x - 10x:

x - 10x = ...999.0 - ...990.0
-9x = 9
x = -1

And while this is true for 10-adic numbers, it is certainly not true for the real numbers.