Repeating Decimals and Rounding I have stumbled across the proof that $0.\overline{9}=1$. The proof is as follows.

Let $x=0.\overline{9}$
$10\cdot x = 9.\overline{9}$
$10\cdot x = 9 + 0.\overline{9}$
Now that $0.\overline{9}=x$, $10\cdot x = 9 + x$.
We get $9x=9$ and $\therefore x=1$.

With this proof, we know that $0.\overline{9}=1$. Is there any scenario in math where using $0.\overline{9}$ instead of $1$ offers an easier solution to a problem?
I don't know of any case where using this representation offers an easier solution to a given problem. But in one case it is useful is Cantor's diagonal argument for the proof of uncountablity of the reals using decimal number system. There you want to make sure that while listing out the reals, that you always consider one representation of such recurring $9$s vs $0$s consistently. Otherwise someone might claim that oh well! the number you constructed is not in our list, but an alternate representation might have it and other more subtle issues.So yeah! without this basic fact, proving uncountability of real becomes harder by Cantor's beautiful agrument.
I don't know of any case where using this representation offers an easier solution to a given problem. But in one case it is useful is Cantor's diagonal argument for the proof of uncountablity of the reals using decimal number system. There you want to make sure that while listing out the reals, that you always consider one representation of such recurring $9$s vs $0$s consistently. Otherwise someone might claim that oh well! the number you constructed is not in our list, but an alternate representation might have it and other more subtle issues.So yeah! without this basic fact, proving uncountability of real becomes harder by Cantor's beautiful agrument.