The following lemma is from Neal Koblitz - p-adic Numbers, p-adic Analysis, and Zeta-Functions-Springer (1996).
Can someone please explain the last part? why does adding a multiple of $p^i$ to the integer $\alpha$ yield an integer between $0$ and $p^i-1$? So , that means that using the fact that $|.|_p$ is a non-Archimedean norm : $|\alpha +tp^i-x|_p\le max(|x-\alpha|_p,|tp^i|_p)\le p^{-i} $ because it was proved that $|x-\alpha|_p\le 1/p^i$ and $|tp^{i}|_p = 1/p^i$. But I don't know how to conclude from this that $0\le \alpha +tp^i \le p^i -1$ nor what the initial interval for $\alpha$ was. Any idea?