Is 0.9999... equal to -1? [closed]

What you have read concerns analysis of functions of $p$-adic numbers, not real ones. Why? Because previous paragraph states that "In the 10-adic numbers, the analogues of decimal expansions run to the left.".

If you think about real numbers as limits of series (here $n$ is an integer): $$x = \sum_{k=n}^\infty \frac{a_k}{10^k} \in \mathbb R, \quad 0 \le a_k < 10$$ then $p$-adic numbers are simply the same series "running to the left". In usual topology (with euclidean metric) it's not convergent, but with $p$-adic metric, it is. $$x = \sum_{k=n}^\infty a_k p^k \in \mathbb Q_p, \quad 0 \le a_k < p.$$