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Proof that 0.999... equals 1

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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

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