Why do eigenvalues of Hermitian/Unitary maps have to be distinct for the eigenvectors to be orthogonal?

Question

Some things I know:

1. Eigenvectors corresponding to DISTINCT eigenvalues are orthogonal for Hermitian and Unitary maps.

2. Hermitian and Unitary maps are normal.

3. Normal maps have an orthogonal basis of eigenvectors.

Surely from this it follows that all eigenvectors of Hermitian and Unitary maps will always be mutually orthogonal, irrespective of whether or not the eigenvalues are the distinct, as we know that the eigenvectors form an orthogonal basis, so by definition of a basis, we can't have any more or fewer eigenvectors that aren't mutually orthogonal with the others.

Edit: I know this isn't true but just wondering where the error in this line of reasoning is. Thanks :)