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.