Let $X$ be a separable, metrizable locally convex space. Suppose $V$ is a neighborhood of $0$ and a barrel (closed, absolutely convex, and absorbing). Show that there exist points $y_n\in X\setminus V$, and open convex neighborhoods $V_n$ of $0$ so that $$ X\setminus V=\bigcup_{n=1}^\infty(y_n+V_n). $$
[THOUGHTS:] The set $y_n+V_n$ is a neighborhood of $y_n$. The subset $V$, by the definition of barrel, is closed, absolutely convex and absorbing. How can I show the existence of $y_n$ and $V_n$ with the desired properties? I'm wondering it should be somehow related to the assumption that $X$ is separable.
[MOTIVATION:] I'm trying to understand the proof of a theorem in locally convex space from a note on functional analysis. The statement above is a step in the proof.
Seperability and metriziability play a role. Note that every subset of a seperable metric space is second countable and hence Lindelöf in the subspace topology. Hence, so is $X \setminus V$. Now, given $y \in X \setminus V$, sa $V$ is closed, there is an open convex neighbourhood $V_y$ of zero, such that $y + V_y \subseteq X \setminus V$. Now $\{y + V_y: y \in X \setminus V\}$ covers $X \setminus V$. Now, by Lindelöfness of $X \setminus V$, there is a countable subcover $\{y_n + V_n: n \in \mathbf N\}$.