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:

$0.999 \ldots = 1$

$\blacksquare$