  Exceptional orthogonal polynomials are complete families of orthogonal
polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio
Dur\'an discovered a gap in the original proof of completeness for exceptional
Hermite polynomials, that has propagated to analogous results for other
exceptional families. In this paper we provide an alternative proof that
follows essentially the same arguments, but provides a direct proof of the key
lemma on which the completeness proof is based. This direct proof makes use of
the theory of trivial monodromy potentials developed by Duistermaat and
Gr\"unbaum and Oblomkov.
