Banach Space Closed Subspace

Question

Let $ \mathcal B$ be a Banach Space. Fix $z \in \mathcal B$ with $z \neq 0$. Consider the set

$$A :=\{y-z : y \notin \operatorname{span} \{z\}, y \in \mathcal B\}.$$

Is it true that $\alpha z \notin \overline{A}$ for any $\alpha \in \mathbb{C}$?

I'm looking for a counterexample or a hint about how to prove it, but just a hint.

Answer 1

I will give you a hint. How small can $y$ be? And is $y-z$ ever in the span of $z$?