# 0.999.../Proof from 0.111...

< 0.999...

## Assumptions[edit]

## Proof[edit]

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.