< 0.999...
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.
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