Proving 0.99999999999999 =1 using Geometric series

Exercise:

Show that, $0.\bar{9}=1$ using limit of series

Solution:  we know that $0.\bar{9}=0.9+0.09+0.009+0.0009+....$

$\Leftrightarrow 0.9+0.09+0.009+0.0009+....=9\left( 0.1+0.01+0.001+0.0001+.... \right)=9\sum\limits_{i=1}^{\infty }{\frac{1}{{{\left( 10 \right)}^{i}}}}$

But $0.1+0.01+0.001+.....$ is an Geometric series of ratio $0.1$ and first term is $0.1$

Hence $0.\bar{9}=9\sum\limits_{i=1}^{\infty }{\frac{1}{{{\left( 10 \right)}^{i}}}=9\underset{n\to \infty }{\mathop{\lim }}\,}\sum\limits_{i=1}^{n}{{{\left( \frac{1}{10} \right)}^{i}}=9\underset{n\to \infty }{\mathop{\lim }}\,}\left( 0.1 \right)\frac{1-{{\left( \frac{1}{10} \right)}^{n}}}{1-0.1}=\frac{0.9}{0.9}\underset{n\to \infty }{\mathop{\lim }}\,\left( 1-{{\left( \frac{1}{10} \right)}^{n}} \right)=1$