Writing the Resolvent as an Integral

Question

Let $A$ be a self-adjoint operator on a Hilbert space, and let $z \in \mathbb{C}$ not in the spectrum of $A$, i.e. the resolvent operator $(A - z)^{-1}$ is well-defined (bounded, etc). I want to justify the identity $$ (A - z)^{-1} = i \int_0^\infty e^{-i(A- z)s} ds$$ My thinking is the following. I would like to formally show that $$ i (A- z) \int_0^\infty e^{-i(A- z)s} ds = 1 $$ the identity operator. It seems like simple change of variables but I'm not sure exactly how to carry this out. It looks like, letting $u = e^{-i(A- z)s}$ that the integral should just turn out to be $\int_0^1 du$ and hence equal to 1. I would appreciate any hints/help! Thanks.