This is a question from Conway, a course in Functional Analisys. Background is the Spectral theorem, which states that for a normal operator $N$ there is a unique spectral measure $E$ on the Borel subsets of $\sigma(N)$ such that $N=\int zdE(z).$
I am lost on how to use this for the question: "Show for $N$ normal there is a sequence of invertible normal operators that converges to $N$."
Also this (The set of invertible normal operator is dense in the set of normal operator) relates to my problem, but I am not sure how to use it. Can I just say my sequence exists?
My own attempt: If $N$ itself is invertible, take any sequence $\{\alpha_i\}\subset\mathbb{C}$ that converges to 1. Then $N_i=\alpha_iN$ suffices. Is this true? What happens when $N$ is not invertible? Should I rather look at convergence of the spectral measures instead?
Any help or hint in the right direction will be greatly appreciated.