Let $E/\mathbb{Q}_p$ be an elliptic curve. Then for $n \geq 1$, let $E_n(\mathbb{Q}) = \left\{P \in E(\mathbb{Q}_p) : \dfrac{x(P)}{y(P)} \in p^n \mathbb{Z}_p\right\}$. According to Cassels in Lectures on Elliptic Curves, if $P=(x,y) \in E_1(\mathbb{Q}_p)$ and we set $u(P) = x/y$ then $|u(sx)|_p = |s|_p |u(x)|_p$ where $s$ is an integer.
Why is this the case? I don't understand his "proof". This follows the lemma which states that $|u(x_1 + x_2) - u(x_1) - u(x_2)| \leq \max\{|u(x_1)|^5, |u(x_2)|^5\}$ for $x_1,x_2 \in E_1(\mathbb{Q}_p)$.