To add to what @10chartley said, the proof is also pretty straightforward:
Let: x = 0.999...
Multiply both sides by 10: 10x = 9.999....
Which can be rewritten as: 10x = 9 + 0.999...
And then: 10x = 9 + x
Subtract x from both sides: 9x = 9
Divide both sides by 9: x = 1
Q.E.D.