Characterization of non-unique decimal expansions

Solution 1:

Yes (up to switching $x$ and $y$). Let $N$ be the least integer such that $x_N\ne y_N$. Assume WLOG that $y_N>$. If $y_N>x_N+1$ then since the remaining digits together contribute at most $10^{-N}$ we have $y-x\ge 2\cdot 10^{-N}-10^{-N}>0$, so $x\ne y$. If $x_{N+1}\ne 9$ or $y_{N+1}\ne 0$, then $x_{N+1}-y_{N+1}\le 8$ and since the remaining digits contribute at most $10^{-N-1}$ we have $$y-x\ge 10^{-N}-8\cdot 10^{-N-1}-10^{-N-1}>0$$ \so $x\ne y$. Thus we must have $x_{N+1}=9$ and $y_{N+1}=0$. Continuing in this manner we see that it is true for all remaining digits.