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.

The argument form

$x = \bar{x}$ therefore $x$ is real

is a standard one; generally, nothing more needs to be said.
If you were in a setting where you felt that you need to include a proof of this rather than just using it, a simple way is
$$ \mathrm{im}(x) = \mathrm{im}(\bar{x}) = - \mathrm{im}(x) $$
and thus
$$ 2 \mathrm{im}(x) = 0 $$
$$ \mathrm{im}(x) = 0$$
