I am struggeling with an easy deduction that I cannot see for some reason today myself, hope to get some help on this: Suppose $T$ is a compact self-adjoint operator on a Hilbert space $\mathcal{H}$. Then I understand that $(Tx,x)$ is always real, how can I deduce from this that for $\lambda \in \mathcal{C}$
\begin{equation} |((T-\lambda)x,x)| \leq |\text{Im}(\lambda)||x|^2 \end{equation}
Many thanks !!