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, 

$1.0 -.9 = .1$

$1.00-.99 = .01$

$1.000-.999=.001$, 

$\ldots$

$1.000\ldots -.99999\ldots = .000\ldots = 0$
