Spectral measure of a self-adjoint operator.

Question

I'm dealing with a proof of a theorem that was left for the reader (not a news sadly).
The proposition states the following :
Let A be a self-adjoint operator and $\{\mu_n\}_{n=1}^{N}$ a family of spectral measures. Then $$\sigma(A)=supp\{\mu_n\}_{n=1}^{N}$$ It's quite easy to show that $supp\{\mu_n\}\subseteq\sigma(A)$, thanks to the spectral theorem. But I have problems showing the other way. The idea I had was to prove that $(supp\{\mu_n\})^c\subseteq\rho(A)$ but I don't know what else to do.
I'm open to any other idea, not necessarily the solution.
Thanks you for the consideration and sorry for my bad English.