Upper Bound for inverse selfadjoint operator between Hilbert space

Question

I am not very steady in Functional Analysis. I try to follow a certain proof and do not get the last part.
Theorem: Let $H$ be a complex Hilbert space and $A\in L(H)$ a bounded linear and self adjoint operator. Then $$\forall z \in \mathbb{C}\backslash \mathbb{R} \Rightarrow ||(z-A)^{-1}||_{L(H)} \leq \frac{1}{|\Im(z)|}$$
For the proof by using the self adjointness we can show that the following inequality holds: $$\forall z\in\mathbb{C}\backslash \mathbb{R} , u \in H\quad ||u||\leq \frac{1}{|\Im(z)|}||(z-A)(u)||_{L(H)}$$ How does this Inequality now implies the above theorem? Probably the answer is easy and I am missing something. Thanks in advance

Answer 1

Correct your inequality by putting an absolute value around $\Im(z),$ and apply it to $u=(z-A)^{-1}(v),$ for any $v\in H.$

Answer 2

Take any $v \in H$ and write $v=(z-A)(u)$ or $u=(z-A)^{-1} v$. We get $\|(z-A)^{-1}v\| \leq \frac 1 {\Im z } \|v\|$.