Suppose $P$ is a separating family of seminorms on a vector space $X.$ Let $L$ be the smallest family of seminorms on a vector space $X$ that contains $P$ and is closed under max. Let $\tau_P$ and $\tau_L$ denote the topologies on $X$ generated by $P$ and $L$ respectively. Show that these two topologies coincide.
My attempt:
I have shown that $\tau_L\subseteq \tau_P.$ To show the reverse inclusion:
Let $A \in \tau_P.$ Let $$B=\left\{\bigcap_{(p,n)\in I} V(p,n):I \subseteq P\times \mathbb{N}, I\text{ is finite}\right\}$$ where $$V(p,n)=\left\{x \in X:p(x)<\frac 1n\right\}$$ Then $B$ is a basis for $\tau_P$. Therefore, $$A=\bigcup_{x\in A\\B_x \in B} x+B_x$$
I'm unable to conclude from this that $A\in \tau_L.$ I especially don't understand how "closed under max" helps here.
Edit: As mentioned in the comments by Jochen, $P \subseteq L \implies \tau_P \subseteq \tau_L.$ Then I don't see the importance of $L$ being closed under max.