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?

Often times people who ask this question are not very convinced by a proof.  Since they may not be particularly math inclined, they may feel that a proof is a sort of sleight-of-hand trick, and I find the following intuitive argument (read "don't down-vote me for lack of rigor, lack of rigor is the point") a bit more convincing:
STEP 1)  If $.99...\neq1$, everyone agrees that it must be less than $1$.  Let $\alpha$ denote $.99...$, this mysterious number less than $1$.
STEP 2)  Using a number line, you can convince them that since $\alpha<1$, there must be another number $\beta$ such that $\alpha<\beta<1$.
STEP 3) Since $\alpha<\beta$, one of the digits of $\beta$ must be bigger than the corresponding digit of $\alpha$.
STEP 4)  However it is usually intitively clear that you cannot make any digit of $.99...$ bigger without making the resulting number (ie $\beta$) bigger than $1$.
STEP 5) Thus no such $\beta$ can exist, and thus $.99...$ cannot be less than $1$.
