The definitions I am using are as follows:
A vector space $V$ equipped with a family $P$ of semi-norms such that $\cap_{p\in P}\{x\in V : p(x) = 0\} = \{0\}$ is called a locally convex vector space.
I am concerned with the topology given by the collections of subsets $U$ of $V$ such that for all $x\in U$ there exists $n\geq 1$ and $p_{1}, ... , p_{n}$ in $P$ and $\epsilon_{1}, ..., \epsilon_{n} > 0$ such that
$x\in \{y\in V : p_{i}(x - y) < \epsilon_{i}$ for $i = 1,... n\}\subset U$.
Showing this topology is closed under finite intersection is very straight-forward. But arbitrary unions is eluding me because if I take a family of such sets, say $\{U_{\alpha}\}$, then I don't get a finite subset of seminorms and epsilons to use to show that the resulting union is still open.
How can I finish this?
Each $x$ in the union must be in one of the particular sets $U$ appearing in the union, and you get your $n$, $p_i$s and $\varepsilon_i$s from this $U$.