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.