functional analysis – The normal real-spectrum operator is hermitian.

Consider a normal operator on a complex Hilbert space with a spectrum contained in the actual line. Show that the operator is Hermitian without using the spectral theorem.

What I've tried – As the spectrum is within the real line, the spectrum limit as a subset of complex numbers is the spectrum itself. The spectrum is therefore equal to the spectrum of the approximate point. I have been unable to go farther.