Let $p$ be a prime number. Define the p-adic absolute value function $|\cdot|_{p}$ on $\mathbb{Q}$:
$|x|_{p}=\left\{ \begin{array}{ll} 0 & \text{if }x=0\\ p^{-k} & \text{if }x=p^{k}\frac{m}{n}\text{ and }\gcd(p,mn)=1\end{array}\right.$
where $m,n\in\mathbb{Z\setminus}\{0\}$ and $p\nmid m$ and $p\nmid n$. Show that for $x,y\in\mathbb{Q}$,
$|x+y|_p \leq max\left\{ |x|_{p},|y|_{p}\right\}$
How do I express $|x+y|_p$?