Let $X$ be a Banach space and $W$ be a proper closed subspace. Let $0 < \epsilon < 1$. Since $X/W$ is nonzero, one can find some $v \in X$ such that $\|v + W\| = \epsilon$. But then it is possible to find some $w \in W$ such that $\|v - w\| = 1$?
It seems to me that it is not immediate from the definition of the quotient norm. Thank you very much.
Consider that $W$ is connected as a subspace of $V$ and that $\|v-\cdot\| : V\to\mathbb{R}$ is continuous. Therefore, $S = \{\|v-w\| : w\in W\}$ is connected in $\mathbb{R}$. As $\epsilon = \inf(S)$ and $S$ is unbounded above (easy to see by the reverse triangle inequality), by the connectedness of $S$ we must have $S = (\epsilon, \infty)$ or $[\epsilon, \infty)$, and therefore, $1\in S$. This implies the existence of a $w\in W$ such that $\|v-w\| = 1$.
