0.999.../Proof from 0.111...

Assumptions

• Decimal long division
• Decimal multiplication by a small number

Proof

Using long division, ¹⁄₉ = 0.111…. Multiplying both sides by 9, we have 1 = 9 × 0.111… = 0.999…. Furthermore, because ^x/_x=1, ⁹/₉=1. However, ¹/₉=0.111..., so multiplying the numerator by nine, and thereby multiplying the answer by nine, produces 0.999... in terms of decimals, but, in terms of fractions, a numerator that is equal to the denominator.