Let $V$ be a vector space with a seminorm $\|\cdot\|_s$. Then apparently we can turn $\|\cdot\|_s$ into a norm $\|\cdot\|$ on $V/W$ by defining $\|v + W\| = \inf_{w \in W} \|v + w\|_s$ where $W$ is any closed subspace of $V$.
It's clear to me that if $V_0$ denotes the kernel of the seminorm $\|\cdot\|_s$ then $\|\cdot\|_s$ turns into a norm on $V/V_0$. What is not so intuitive to me is what happens if $W$ is disjoint from $V_0$. I think the fact that the norm $\|\cdot\|$ defined above then is a norm means that for every $v_0 \in V_0$ there is a sequence in $W$ converging to it. This holds because $0 \in W$ hence there is a sequence $w_n$ converging to $0$ and if $v_0 \in V_0$ then $v_0$ is in every neighbourhood of $0$, hence $w_n$ also converges to $v_0$. Is this correct?
If yes, would you show me some concrete examples illustrating this to help me develop some intuition? Thanks.