Let $x \in \mathbb{Q}$. Show that if $\alpha \in \mathbb{Z}$ is an integer such that $|\alpha - x|_{p} \leq p^{-i} $ for some $i\in \mathbb{N}$, then there exists $\alpha' \in \{ 0,1,2, \ldots, p^{i}-1 \}$ such that $|\alpha' - x|_{p} \leq p^{-i} $.
Could you give me a little hint to approach this problem?