Is it true that $0.999999999\ldots=1$? I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?

\begin{align}
x &= 0.999... \\
10x &= 9.999... \\
&= 9 + 0.999... \\
&= 9 + x \\
10x - x &= (9 + x) - x \\
(10 - 1)x &= 9 + (x - x) \\
9x &= 9 \\
x &= 1
\end{align}
