0.999...=1/Proof 2

Theorem

$0.999 \ldots = 1$

Proof

\(\ds 0.333 \ldots\)

\(=\)

\(\ds 1 / 3\)

\(\ds \leadsto \ \ \)

\(\ds 3 \paren {0.333 \ldots}\)

\(=\)

\(\ds 3 \paren {1 / 3}\)

\(\ds \leadsto \ \ \)

\(\ds 0.999 \ldots\)

\(=\)

\(\ds 3 / 3\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$