For every bounded normal operator $N \in \mathcal{B(H)}$, and every $\epsilon>0$, there is a set $\{P_n\}$ of pairwise commuting orthogonal projections with sum $I$(identity operator) and corresponding set $\{\lambda_n\}$ in $\mathbb{C}$ s.t $||N-\sum_{n}\lambda_nP_n|| < \epsilon$.
I was thinking using functional calculus for normal operator to prove it, but I am still unclear how to start it. Furthermore, there is another similar problem: for a bounded normal operator T such that $0\leq T\leq I$, find a sequence of pairwise commuting projections s.t $T=\sum_n \frac{1}{2^n}P_n$, there is hint for this problem: Let $P_1=\chi_{(1/2,1]}, P_2=\chi_{(1/4,1/2]\cup(3/4,1]}, \dots$. I am not sure how to solve those two problems and their connections.
Any help will be appreciated.
Using compactness of $\sigma(N)$, find a partition $\sigma(N)=\bigcup_{n=1}^k K_n$ with each $K_n$ measurable and contained in a ball of radius $\varepsilon$. Now let $P_n=1_{K_n}(N)$, and $\lambda_n\in K_n$. Then \begin{align} \|N-\sum_n\lambda_nP_n\|&=\left\|\sum_n (NP_n-\lambda_nP_n)\right\| =\max_n\|NP_n-\lambda_nP_n\| \\ &=\max_n\left\|\int_{K_n}(\lambda-\lambda_n)\,dE(\lambda) \right\| \leq\max_n\varepsilon\|P_n\|=\varepsilon. \end{align}
For the second part, you use the identity $$ t=\sum_{n=1}^\infty 2^{-n}\,1_{R_n}(t),=\sum_{n=1}^n2^{-n}\,1_{\{t\geq 1-2^{-n}\}}(t). $$ where $R_n=\{t:\ t\geq\sum_{k=1}^n2^{-k}\}$.