Let $\|f\|_{n, m} = \sum_{i<n}(|i|^m+10)|a_i|$ denote a family of seminorms on $\mathbb{C}(t)$. Note that for any $\{(n_v, m_v)\}_{v\text{ in finite subset of }\mathbb{N}}$ there is a seminorm $\|.\|_{n,m}$ such that $\|f\|_{n, m}\geq\sup_v\|f\|_{n_v, m_v}$. Also these seminorms define the same topology as stated in the problem. Resulting topological space is separabel so for some countable set of $F_i\in\mathbb{C}(t)$ this set is dense. Let $\{U_{\lambda}\}$ be some open cover of this topological space. For each $F_i$ and $(n, m)$ define $\varepsilon_{i, n, m}$ as supremum of $\varepsilon >0$ such that the $\varepsilon$ ball around $F_i$ in $\|.\|_{n, m}$ lies inside of $U_{\lambda}$ for some $\lambda$ and let $\{\|f\|_{n, m}<\varepsilon_{i, n, m}/2\}\subset U_{\lambda_{i, n, m}}$ in result $\cup_{i, n, m}U_{\lambda_{i, n, m}}$ is countable open subcover.

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