Given an unbounded self-adjoint operator $A$ on some Hilbert space $\mathcal{H}$, and $\mu$ a non zero real number: I want to show that \begin{equation} \lVert (\mathbf{A} + i\mu \mathbf{I})^{-1} \rVert \leq \frac{1}{|\mu|} \end{equation}
I have tried to compute \begin{align} \left( (\mathbf{A}+i\mu \mathbf{I})^{-1} \psi, \psi \right) &= \left( (\mathbf{A}+i\mu \mathbf{I})^{-1} \psi, (\mathbf{A}+i\mu \mathbf{I})(\mathbf{A}+i\mu \mathbf{I})^{-1} \psi \right) \\ &= \left( \mathbf{A}(\mathbf{A}+i\mu \mathbf{I})^{-1} \psi, (\mathbf{A}+i\mu \mathbf{I})^{-1} \psi \right) + i\mu {\lVert (\mathbf{A} + i\mu \mathbf{I})^{-1} \psi \rVert}^2 \end{align}
Unfortunately by this point I have no idea how to go further, I've tried playing with inequalities but I got to nothing. I'd really appreciate any suggestion.