0.999... = 1

however

0.99999 <> 1

The repeating decimal is identical to 1, whereas the fixed decimal is not.

let x = 0.999... (where the ellipsis denotes infinite repetition of '9')

so 10x = 9.999...

Clearly, 10x - x = 9x

10x = 9.999...
- x = - 0.999...
------ ----------
9x = 9

Dividing both sides by 9 gives x=1.

Therefore 0.999... is identically equal to 1, mathematically, theoretically, and, in fact. (Well, that's what I think anyway.)

k