0.999...=1/Proof 2

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Theorem

$0.999 \ldots = 1$


Proof

\(\displaystyle 0.333 \ldots\) \(=\) \(\displaystyle 1 / 3\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 3 \paren {0.333 \ldots}\) \(=\) \(\displaystyle 3 \paren {1 / 3}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0.999 \ldots\) \(=\) \(\displaystyle 3 / 3\)
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$