0.999.../Proof from 0.333...

Assumptions

• Decimal long division
• Decimal multiplication by a small number

Proof

Using long division,

$\frac{1}{3}=\mathrm{0.333...}$

Multiplying both sides by 3, we have

$\frac{3}{3}=\mathrm{0.999...}$

and therefore...

$1=\mathrm{0.999...}$