I almost have a homework problem solved but I've used a claim that might be dubious.
The setting is this: Let $(V,Q)$ be a locally convex space ($Q$ is the family of seminorms inducing the topology on $V$). And let $q\in Q$.
Claim:
For any neighborhood $U$ of $0$ in $V$. There exists a sufficiently small $\epsilon > 0$ such that $q^{-1}([0,\epsilon))\subset U$.
The reason this claim helps me is that I need to prove something about all neighborhoods of $0$ in $V$, and it greatly simplifies things if I can simplify my situation to sets of the form $q^{-1}([0,\epsilon))$.