I'm treating the fact that we know eigenvectors of symmetric matrices corresponding to distinct eigenvalues must be orthogonal as a given and trying to show (real) symmetric matrices are diagonalizable.
Knowing the aforementioned fact, we can conclude that there exists an orthogonal basis of eigenvectors for any symmetric matrix. We know a matrix A is diagonalizable iff there exists a basis of eigenvectors. So therefore, symmetric matrices are diagonalizable. ∎
I'm not sure if I'm making a bit of a logical leap when I can conclude that there exists an orthogonal basis of eigenvectors, would appreciate any criticism/advice.