Many numbers can be symbolically written in more than one way. For example :

$1=\frac{1}{1}=\frac{2}{2}=\frac{3}{3}=\cdots$

The decimal point notation also has its own problems. For example, consider the following perfectly valid identity :

$1=0.99999999\cdots~,$

with the series of $9$s repeating ad infinitum. There are many ways to prove the above. Here is one.

The right-hand side can be written in a form involving a geometric progression :

$9 \sum_{n=1}^{\infty} 10^{-n}~.$

The series converges in $\mathbb{R}$. Its value can thus be computed using the usual formula

$\sum_{n=1}^{k} r^n=\frac{r~(1-r^{k})}{1-r}~.$

with $0\leq r<1$, $r^n\rightarrow 0$, which, in our case gives

$\sum_{n=1}^{\infty} 10^{-n}=\frac{1}{9}~.$

Which proves the identity.