# 0.999...=1/Proof 3

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## 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:

- $0.999 \ldots = 1$

$\blacksquare$