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

The decimal point notation also has its own problems. It is common knowledge that

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

The right-hand side can be written:

\begin{equation} 9 \sum_{n=1}^{\infty} 10^{-n}~. \end{equation} The series converges in $\mathbb{R}$ and is just the geometric series of reason $\frac{1}{10}$. Its value can thus be computed using the usual formula

\begin{equation} \sum_{n=0}^{k-1} r^n=\frac{1-r^k}{1-r}~. \end{equation} with $0\leq r<1$, $r^n\rightarrow 0$ and \begin{equation} \sum_{n=1}^{\infty} 10^{-n}=\frac{1}{9}~. \end{equation} Which proves the identity. In the end, this is just a matter of playing around with infinity.


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