Look at the last sentence of ยง22.3 on page 535. We have:

For any

,

, and operator

.

Let us take

, and let

be the hermitian operator

, i.e.

. Then the above becomes:

i.e.

If

is an eigenfunction of

with eigenvalue

, then

Since

is always a real number, that means that the eigenvalue

must always be real, as well.