0.999...=1/Proof 3
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Theorem
- $0.999 \ldots = 1$
Proof
Let $c = 0.999 \ldots$
Then:
\(\ds c\) | \(=\) | \(\ds 0.999 \ldots\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 10 c\) | \(=\) | \(\ds \paren {9.999 \ldots}\) | multiplying $c$ by $10$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 10 c - c\) | \(=\) | \(\ds \paren {9.999 \ldots} - \paren {0.999 \ldots}\) | subtracting $c$ from each side | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 9 c\) | \(=\) | \(\ds 9\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c\) | \(=\) | \(\ds 1\) |
It follows that:
- $0.999 \ldots = 1$
$\blacksquare$