Let $\mathcal D(\mathbb{R})$ be the space of test functions. Let $$p_{\{a_k\}}(\varphi)=\sum_{k=-\infty}^\infty a_k \max_{m=0,...,a_k}\max_{x\in[k, k+1]}|\varphi^{(m)}(x)|$$ where $\{a_k\}$ is the sequence of non-negative integers. This defines seminorms on $\mathcal D(\mathbb{R})$. $\mathcal D'(\mathbb{R})$ is the space of continuous functionals with respect to this seminorms.
Let $\{F_j\}$ be a sequence of $\mathcal D'(\mathbb{R})$ such that $F_j(\varphi)$ tends to $0$ for every $\varphi \in \mathcal D(\mathbb{R})$. How to prove there exists a sequence $\{a_k\}$ of non-negative integers and numbers $\varepsilon_j \rightarrow 0$ such that $$|F_j(\varphi)|\leqslant \varepsilon_j p_{\{a_k\}}(\varphi)$$ for every j?
P.S. I can prove the existence of seminorm $p_{\{a_k\}}$ such that $$|F_j(\varphi)|\leqslant C_j p_{\{a_k\}}(\varphi)$$ but can not prove that exists sequence $\varepsilon_j \rightarrow 0$ sutisfying the condition.