Equivalent conditions for the ramification group(s) in a finite Galois extension

Equivalent conditions for the ramification group(s) in a finite Galois
extension

Let $L|K$ be a finite Galois extension and $v$ a discrete normalized
valuation on $L$ such that its restriction to $K$ extends uniquely to $L$.
(1) Why is $G_1=\{\sigma\in G(L|K)\mid v(\sigma(x)-x)\ge 2\ \ \forall
x\in\mathcal{O}\}$ equal to the ramification group $R(L|K)=\{\sigma\in
G(L|K)\mid v\left(\frac{\sigma(x)}{x}-1\right)>0\ \ \forall x\in L^*\}$?
(Here $\mathcal{O}$ is the valuation ring of $L$)
(2) Is it true that in general $G_s$ equals $\{\sigma\in G(L|K)\mid
v(\sigma(\pi)-\pi)\ge s+1\}$, where $\pi$ is any element of $L^*$ such
that $v(\pi)=1$? (The $G_s$'s are the higher ramification groups).
This question arises from Chapter II, $\S 10$ of Neukirch's Algebraic
Number Theory.