x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999... = 9
-> 9x = 9
-> x = 1 thus, 0.999... = 1

1 - 0.999999... = ε              (0)

9 × (1 - 0.999999...) = ε × 9

0.999... ×
9        =
───────────
8.1
  81
   81
     .
      .
       .

───────────
8.999...

9 - 8.999999... = 9ε              (1)

8.999999... = 8 + 0.99999...  (2)

8.00000000... +
0.99999999... =
────────────────
8.99999999...

9 - 8.999999... = 9ε                      because of (2)
9 - 8.999999... = 9 - (8 + 0.99999...) =  because of (1)
                = 9 -  8 - (1 - ε)        because of (0)
                =   1    -  1 + ε
                =               ε.

9ε = ε

8ε = 0

ε = 0

1 - 0.999999... = ε = 0

1 / 3 = 0.333... (by long division)

=> 0.333 * 3 = 0.999... (multiplying each digit by 3)

But then we already know 0.333... * 3 = 1

Therefore 0.999... = 1