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Fraction proof

1. $0.333... = \frac{1}{3}$
2. $0.999... = 1$ (multiply both sides by 3)

QED

Algebra proof

1. $x = 0.999...$
2. $10x = 9.999...$ (Multiply both sides by 10)
3. $10x - x = 9.999... - .999...$ (Subtract x from both sides)
4. $9x = 9$ (Simplify)
5. $x = 1$ (Divide both sides by 9)

QED