Conditions for locally convex space to be normable

Question

Let $X$ be a Hausdorff locally convex space and $P$ is a family of seminorms on $X$. How to show that $X$ is normable iff $P$ is equivalent to a finite subfamily $P_0 \subset P$? One implication is obvious, we can take as a norm maximum of subfamily $P_0$, but another implication isn't so trivial.

Answer 1

So, other implication is also trivial, suppose that $X$ is normable denote by p the norm, then $\exists p_1, \ldots p_n$ - seminorms, such that $$ p \leq Cmax_kp_k$$ It's obvious that topology $\tau(p_1 \ldots p_n)$ is equivalent to the given topology, because for $p_j \in P$ $$p_j \leq C_j p \leq C_jCmax_kp_k$$