= Solution
View a possible complex <eigenvector> $v$ using the <Hermitian inner product>. For a real <symmetric matrix>, $A^\dagger=A$, so $v^\dagger Av$ is real. Since $v^\dagger Av=\lambda v^\dagger v$ and $v^\dagger v>0$, \b[every <eigenvalue> is real].
For <eigenvectors> $v,w$ with distinct <eigenvalues> $\lambda,\mu$, symmetry gives
$$
\lambda v^\dagger w=(Av)^\dagger w=v^\dagger Aw=\mu v^\dagger w,
$$
so \b[$v^\dagger w=0$]. In particular, real <eigenvectors> are <orthogonal> in the real <dot product>.
Use the permitted diagonalizability assumption to take bases of the real <eigenspaces> whose union spans $\mathbb R^n$. Apply the <Gram-Schmidt process> within each <eigenspace>. Its linear combinations remain <eigenvectors> of that same <eigenvalue>; vectors in different <eigenspaces> are already <orthogonal>. The resulting union is \b[an <orthonormal basis> of <eigenvectors>], giving an <orthogonal diagonalization of a real symmetric matrix>.
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