We consider the following two families of seminorms on the Schwartz space and want to show that they induce the same topology.
First family: $(\| \cdot \|_N) $ for $N \in \mathbb{N}$ defined by $$ \| f \|_N=\sup_{\vert \alpha \vert \leq N} \sup_{x\in \mathbb{R}^n} (1+\vert x \vert ^2)^N \vert (\partial^\alpha f)(x) \vert. $$
Second family: $(\| \cdot \|_{(K,\alpha)})$ for $K \in \mathbb{N}$ and $\alpha$ a multi-index defined by $$ \| f \|_{(K,\alpha)}=\sup_{x \in \mathbb{R}^n} (1+\vert x \vert)^K \vert (\partial^\alpha f)(x)\vert. $$
I am not very familiar with Frechet spaces but from my topological knowledge I suspect that it is sufficient to show $$ \| f \|_N \leq C \sum_{i=1}^l \| f \|_{(K_i,\alpha_i)} $$ for some for each $N \in \mathbb{N}$, $C>0$ and $l,K_i\in \mathbb{N}$ and some multi-indices $\alpha_i$ where $C,l,K_i,\alpha_i$ may depend on $N$ and
$$ \| f \|_{(K,\alpha)} \leq C \sum_{i=1}^l \| f \|_{N_i} $$
for each $K \in \mathbb{N}$ and each multi-index $\alpha$ for some $C>0$, some $l \in \mathbb{N}$ and $N_i \in \mathbb{N}$ where $C,l,N_i$ may depend on $K$ and $\alpha$.
Is this approach correct or do I need to change something?