in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3):
If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda dE_\lambda$, then for every polynomial $p(\lambda)$ we have $$p(A)=\int_m^{M+\varepsilon} p(\lambda) dE_\lambda.$$ Proof goes like this: $$\left\|\left(A-\sum_{k=1}^n \lambda_k E(\Delta_k)\right)^2\right\| \le \eta^2.$$
Then author says that "Using the relations $E(\Delta_k)E(\Delta_j) = 0$ if $k\neq j$ and $\|AS-A^2\| \le \|A\| \eta$ we find that $$\left\|A^2-\sum_{k=1}^n \lambda^2_k E(\Delta_k)\right\| \le c\eta."$$
Maybe I'm too tired today, but why is the first relation true and how to use both to get the last inequality?
($E(\Delta_k) = E_{\lambda_k} - E_{\lambda_{k-1}}$ and $m = \lambda_0 < \lambda_1 < \ldots <\lambda_n = M+\varepsilon$ is the partition of $[m,M+\varepsilon]$ )
Thanks!