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?
What does it mean when you refer to $.99999\ldots$?  Symbols don't mean anything in particular until you've defined what you mean by them.

In this case the definition is that you are taking the limit of $.9$, $.99$, $.999$, $.9999$, etc.  What does it mean to say that limit is $1$?  Well, it means that no matter how small a number $x$ you pick, I can show you a point in that sequence such that all further numbers in the sequence are within distance $x$ of $1$.  But certainly whatever number you choose your number is bigger than $10^{-k}$ for some $k$.  So I can just pick my point to be the $k$th spot in the sequence.

A more intuitive way of explaining the above argument is that the reason $.99999\ldots = 1$ is that their difference is zero.  So let's subtract $1.0000\ldots -.99999\ldots = .00000\ldots = 0$.  That is, 

$1.0 -.9 = .1$

$1.00-.99 = .01$

$1.000-.999=.001$, 

$\ldots$

$1.000\ldots -.99999\ldots = .000\ldots = 0$
The more 9's there are in the sequence, the smaller the margin between 1 and that number becomes, and after every 9, the margin become 10 time smaller. For example, for the number 0.9, it is ${1 \over \mathrm{10}^{1}}$ away from reaching 1, 0.99 is ${1 \over \mathrm{10}^{2}}$ away from reaching 1, and 0.999 is ${1 \over \mathrm{10}^{3}}$ away from reaching 1. So if $0.99999999 \dots$ has an $n$ number of 9's in it, the 'spacing' between that number and 1 is ${1 \over \mathrm{10}^{n}}$. Now, since $0.99999999 \dots$ has an infinite number of 9's, it makes sense that the 'space' between $0.9999999 \dots$ and 1 is     ${1 \over \mathrm{10}^{\infty}}$, or $\mathrm{10}^{-\infty}$, which is 0. Now, to prove that $\mathrm{10}^{-\infty}$ is 0. Simply take the log base 10 of both sides, and you'll find that $ \log_{10} 0 = -\infty$. This means that the 'spacing' between $ 0.9999\dots $ and 1 is 0, meaning that $0.99999\dots + 0 = 1$.  

Therefore, $0.9999\dots = 1$