$10$ adic numbers Does infinite equal $-1$

Question

Sorry math newbie here. I saw a video that said $9999$ repeating actually equals $-1$. This logic helped people understand $10$ adic numbers.

Under this logic doesn't that mean infinity is equal to $-1$?

Answer 1

"actually equals". N.B. I use $\lim_{N\to\infty}\sum_{n=0}^N9\cdot10^n$ as a precise interpretation of "$99999\cdots$ repeating". Notice $\sum_{n=0}^N9\cdot10^n=\underset{N+1\text{ times }}{\underbrace{99999\cdots999}}$.

$\lim_{N\to\infty}\sum_{n=0}^N9\cdot 10^n=-1$ in the $10$-adics, sure. And $\lim_{N\to\infty}\sum_{n=0}^N9\cdot10^n=+\infty$ in the extended real numbers. We've taken limits in different spaces and arrived at different answers.

Nowhere in the above is there any logical link which deduces $+\infty=-1$ as an equality of extended reals; it is false. There are just different rules at play.