Lemma 13.6.2.label Let $E$ be a vector space over $K \in \RC$, $\rho: E \to [0, \infty)$ be a seminorm, $M \subset E$ be a vector subspace, $\rho_{M}: E/M \to [0, \infty)$ be its quotient by $M$, and $x \in E$ with $\rho_{M}(x) < 1$, then there exists $x_{0} \in B_{E}(0, 1)$ such that $x + M = x_{0} + M$.
Proof. Since $\inf_{y \in M}\rho(x - y) = \rho_{M}(x) < 1$, there exists $y \in M$ with $\rho(x - y) < 1$. Let $x_{0} = x - y$, then $\rho(x_{0}) < 1$ as well and $x_{0} + M = (x - y) + M = x + M$.$\square$
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