Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators".

The motivating problem for this is to find an approximate kernel for the inverse of a differential operator, and as part of this the convergence of a certain series, defined below in $(1)$, is needed.

Let $\Omega$ denote an open subset of $\mathbb{R}^n$ and suppose we have a sequence $k_j(x,\xi)$ of $C^\infty$ functions in $\Omega \times (\mathbb{R}^n \setminus \{0\})$, homogeneous of degree $-(m+j)$ with respect to $\xi$ (here $m$ is a positive integer).

We note this means that, if $K$ is any compact subset of $\Omega$ and $\alpha,\beta$ are any multi - indices then there exists a constant $C^{(j)}_{K,\alpha,\beta}$ so that $$\tag{2} \left|\partial^\alpha_\xi\partial^\beta_x k_j(x,\xi)\right| < C^{(j)}_{K,\alpha,\beta}\left|\xi\right|^{(-(m+j+|\alpha|)}\,, \qquad \forall x \in K, \xi \in (\mathbb{R}^n \setminus \{0\})\,.$$

further, let $\chi(t)$ be a $C^\infty$ function on $\mathbb{R}$ such that $\chi(t) = 0$ whenever $t < 1/2$ and $\chi(t) = 1$ for all $t > 1$. choose a sequence $\{\rho_j\}$ of positive numbers so that $\rho_j \to \infty$ as $j \to \infty$. for each non-negative integer $j$, set $$\chi_j(\xi) = \chi(\rho_j^{-1}\left|\xi\right|)\,.$$

now the claim is that the series $$\tag{1} \sum^\infty_{j = 0} \chi_j(\xi)k_j(x,\xi)$$ converges in $C^\infty(\Omega \times (\mathbb{R}^n \setminus \{0\}))$.

The argument given in the book goes as follows: Choose an exhausting sequence of compact subsets $K_v$, $v = 1,2,\dots$ of $\Omega$, so $$\Omega = \bigcup_v K_v \qquad \text{and} \qquad \text{for each } v \colon K_v \subsetneq K_{v+1} \,.$$ We then use equation $(2)$ together with the facts that, on the support of $\chi_j$ we have $\left|\xi\right| \geq \rho_j/2$ whilst on the support of the gradient $d\chi_j$ we have $\left|\xi\right| \leq \rho_j$, to derive (possibly after suitably adjusting the constants ($C^{(j)}_{K_v,\alpha,\beta}$) that $$\tag{3} \left|\partial^\alpha_\xi \partial^\beta_x \sum^{\infty}_{j = 0} \chi_j(\xi) k_j(x,\xi) \right| \leq \left|\xi\right|^{-(m+|\alpha|)} \sum^\infty_{j = 0} C^{(j)}_{K_v,\alpha,\beta}\rho^{-j}_j$$ It suffices then to require $$\tag{4} \rho_j \geq 2 \sup_{v \leq j,|\alpha + \beta| \leq j} C^{(j)}_{K_v,\alpha,\beta}\,^{1/j}$$ in order to reach the conclusion that the series converges in $C^\infty(\Omega \times (\mathbb{R}^n \setminus \{0\}))$.

$\textbf{Question 1:}$ How do we get to the inequality $(3)$? I tried hard to figure out the details, (happy to provide my calculations in case it helps) but somehow my estimate always turns out wrong ..

$\bf{Question 2:}$ why is the condition that we set on the $\rho_j$ in $(4)$ enough to prove convergence ?