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?

Another approach is the following:
$$\begin{align}
0.\overline9 &=\lim_{n \to \infty} 0.\underbrace{99\dots 9}_{n\text{ times}} \\
&= \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{9}{10^k} \\
&=\lim_{n \to \infty} 1-\frac{1}{10^n} \\
&=1-\lim_{n \to \infty} \frac{1}{10^n} \\
&=1.
\end{align}$$ 
