Topologies generated by the two family of seminorms coincide 
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.
 A: Here I use slightly different symbols for those two sets of seminorms, which is from an exercise in Rudin's book.  Let $  \mathscr{Q} $ be the smallest family of seminorms on a vector space $ X $ that contains $  \mathscr{P} $ and is closed under max. 
Firstly, by the usual technique in proving equality of two sets it's not hard to show:
    $$ \mathscr{Q}=\{\max_{p \in P}p: P\text { a finite subset of } \mathscr{P}\} $$
     Associate to each $p \in \mathscr{P}$ and to each positive integer $ n$ the set $$ V(p, n)=\left\{x: p(x)<\frac{1}{n}\right\} $$
     Then $\tau_{\mathscr{P}}$ is the collection of all finite intersections of the sets $V(p, n)$. 
Now we compare  $\tau_{\mathscr{P}}$ and  $\tau_{\mathscr{Q}}$. 
On the one hand, $\tau_{\mathscr{P}} \subset \tau_{\mathscr{Q}} $ since $ \mathscr{P} \subset \mathscr{Q} $. On the other hand, $ \forall q \in \mathscr{Q}$, using the fact that $ q $ is the max of finite number of seminorms in $ \mathscr{P} $, we conculude:
    $$ V(q, n)=\bigcap_{p \in P}\{x: p(x)<\frac{1}{n}\} $$
    where $ P $ is a finite subset of $ \mathscr{P} $ such that $ q = \max_{p \in P}p $. Thus we have shown  $\tau_{\mathscr{Q}} \subset \tau_{\mathscr{P}} $.
