Let $F$ be a non-Archimedean local field, and $N$ the subgroup of $G=GL_2(F)$ of the form $\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$ with $x \in F$. Let $(\pi,V)$ be a smooth representation of $N$, and $\mu_N$ a Haar measure on N. Consider the linear subspace $V(N)$ of $V$ generated by vectors of the form $\pi(n)v-v$ with $n \in N, v \in V$.
My question concerns the following equivalence: a vector $v \in V$ is also in $V(N)$ if and only if there exists a compact open subgroup $N_0$ of $N$ such that \begin{equation} \int_{N_0}\pi(n)v\, d\mu_N(n)=0. \label{eq1} \end{equation}
The textbook (The Local Langlands for GL(2) - Bushnell, Henniart) I'm using offers the following proof:
Recall $N \cong F$ as groups and $F$ is the union of an ascending sequence of compact open subgroups. Now assume $v \in V(N)$, that is, there exists $n_1,\dots,n_r \in N$ such that \begin{equation*} \sum_{i=1}^{r} \pi(n_i)v_i-v_i=v. \end{equation*} By the above, we can find a compact open subgroup $N_0$ of $N$ that contains $n_1,\dots,n_r$. The above integral is zero for this choice of $N_0$.
For the converse direction, let $v \in V$ and suppose there is a compact open subgroup $N_0$ of $N$ such that the above integral equality holds. Then there exists a normal open subgroup $N_1$ of $N_0$ with $v \in V^{N_1}=\{v \in V:\pi(n)v=v\, \forall n \in N_1 \}$. The space $V^{N_1}$ carries a representation of the finite group $N_0/N_1$. Therefore, \begin{equation} V^{N_1}=V^{N_1}(N_0/N_1) \oplus V^{N_0} \end{equation} and the map \begin{equation} \omega \mapsto \mu_N(N_0)^{-1} \int_{N_0}\pi(n)v\, d\mu_N(n) \end{equation} is the $N_0$-projection $N_1 \rightarrow N_0$. This has kernel $V^{N_1}(N_0/N_1) \subset V(N)$ so we are done.
My questions:
(1) When they claim $F$ is the union of an ascending sequence of compact open subgroups do they explicitly mean that $F=\bigcup_{i \geq 1} p^{-i}\mathfrak{o}$ where $\mathfrak{o}$ is the discrete valuation ring of $F$, and $p$ is the characteristic of $\mathfrak{o}/\mathfrak{p}$ ($\mathfrak{p}$ being the maximal ideal of $\mathfrak{o}$)?
(2) After choosing $N_0$ as above, I compute the integral to the following point: \begin{align*} \int_{N_0}\pi(n)v\, d\mu_N(n)&= \int_{N_0}\pi(n)(\sum_{i=1}^{r} \pi(n_i)v_i-v_i)\, d\mu_N(n) \\ &= \sum_{i=1}^{r} \int_{N_0} (\pi(nn_i)v_i-\pi(n)v_i)\, d\mu_N(n). \end{align*} It's not clear to me why this is zero.
(3) In the converse direction of the proof, I don't see why $N_1$ should exist with $v \in V^{N_1}$.