A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$
$$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$
$$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$
Identity:
- $\mathbb{Z}_p=\{ x \in \mathbb{Q}_p | |x|_p \leq 1 \}$
Could you help me to prove the above identity?
EDIT:
When $x \in Z_p$, it means that: $$x=a_0+a_1p+a_2p^2+ \dots$$ $a_j \in \{0,1, \dots, p-1 \}$ $$$$ So that $x \in Q_p: x=\frac{r}{s}, r,s \in Z_p, s \neq 0 \Rightarrow x=\frac{b_0+b_1 p+b_2 p^2+ \dots}{c_0+c_1p+c_2p^2+ \dots}$
$$x=a_0+a_1p+a_2p^2+ \dots=\frac{a_0+a_1p+a_2p^2+ \dots}{1}$$
So, $x \in Q_p$.
Is it right so far? Also, how can I show that $|x|_p \leq 1$ ?