Eigenvalue-related statements

Question

How can I prove that the following statements are equivalent?

1. $\lambda$ is an eigenvalue of $A+\delta A$, where $\|\delta A\|_{2}\leq \epsilon$
2. $\exists u\in \mathbb{C}^{m}$ such that $\|(A-\lambda I)u\|_{2}\leq\epsilon$ and $\|u\|_{2}=1$
3. $\sigma_{n}(\lambda I - A)\leq \epsilon$, where $\sigma_{n}$ is the smallest singular value of A
4. $\|(\lambda A - I)^{-1}\|\geq \epsilon^{-1}$

I somehow managed to prove 4 implies 1 and 1 implies 2 (even though I'm not certain it's correct), but I am really confused about the remaining ones (2 implies 3 and 3 implies 4). Any tips or help would be greatly appreciated.

Answer 1

I have received a reply on MathOverflow from Denis Serre that might be helpful to anybody having this question:

"This is rather classical. The set of such λ's is called the ϵ-pseudo-spectrum. It is presented in many books on numerical linear algebra. I suggest S.K. Godunov Modern aspects of linear algebra, AMS (1998)."