Let $p$ be a prime number. Define what is meant by the $p$-adic metric $d_{p}$ on $\mathbb{Q}$. Show that for $a, b, c \in \mathbb{Q}$ we have

$$d_{p}(a, b) \leqslant \max \left\{d_{p}(a, c), d_{p}(c, b)\right\}$$

Show that the sequence $\left(a_{n}\right)$, where $a_{n}=1+p+\cdots+p^{n-1}$, converges to some element in (D.

For $a \in \mathbb{Q}$ define $|a|_{p}=d_{p}(a, 0)$. Show that if $a, b \in \mathbb{Q}$ and if $|a|_{p} \neq|b|_{p}$, then

$$|a+b|_{p}=\max \left\{|a|_{p},|b|_{p}\right\} .$$

Let $a \in \mathbb{Q}$ and let $B(a, \delta)$ be the open ball with centre $a$ and radius $\delta>0$, with respect to the metric $d_{p}$. Show that $B(a, \delta)$ is a closed subset of $\mathbb{Q}$ with respect to the topology induced by $d_{p}$.