Let $A$ be a linear operator on Hilbert space $H$. We say that $\lambda \in \mathbb{C}$ is in approximative spectrum of $A$ iff there exists a sequence $(x_n)$ of vectors such that $\|x_n\|=1$ and $\|Ax_n-\lambda x_n\|\rightarrow 0$. Equivalently, $\lambda$ is in approximative spectrum of $A$ iff for every $\varepsilon >0$ there exists $x \neq 0$ such that $\|Ax-\lambda x\| \leq \varepsilon \|x\|$.
How to prove that approximative spectrum is a closed subset of $\mathbb{C}$ ?
Thanks.