How do I prove, using Dirac notation, that $\langle A|A\rangle$ is a real number? Using bra-ket notation, how do I prove that the inner product of a ket with itself (ie, $\langle A | A \rangle$) is a real number?
I understand that the rule of inner products states that: $\langle B | A \rangle = \overline{\langle A | B \rangle}$.
Therefore $|A\rangle$ is equal to it's own complex conjugate, and hence must have no imaginary part, but I'm having trouble writing the proof in a formal way.

If you want/need more detail you could set $\langle A|A \rangle = x + iy$ and then use $\overline{\langle A|A \rangle} = \langle A|A \rangle$ to conclude that $-y = y$, which is true only for $y=0$.
