# 0.999...=1/Proof 3

## Theorem

$0.999 \ldots = 1$

## Proof

Let $c = 0.999 \ldots$

Then:

 $\displaystyle c$ $=$ $\displaystyle 0.999 \ldots$ $\displaystyle \leadsto \ \$ $\displaystyle 10 c$ $=$ $\displaystyle \paren {9.999 \ldots}$ multiplying $c$ by $10$ $\displaystyle \leadsto \ \$ $\displaystyle 10 c - c$ $=$ $\displaystyle \paren {9.999 \ldots} - \paren {0.999 \ldots}$ subtracting $c$ from each side $\displaystyle \leadsto \ \$ $\displaystyle 9 c$ $=$ $\displaystyle 9$ $\displaystyle \leadsto \ \$ $\displaystyle c$ $=$ $\displaystyle 1$

It follows that:

$0.999 \ldots = 1$

$\blacksquare$