  We consider the first family of $\Hcurl$-conforming Ned\'el\'ec finite
elements on tetrahedral meshes. Spectral approximation ($p$-version) is
achieved by keeping the mesh fixed and raising the polynomial degree $p$
uniformly in all mesh cells. We prove that the associated subspaces of
discretely weakly divergence free piecewise polynomial vector fields enjoy a
long conjectured discrete compactness property as $p\to\infty$. This permits us
to conclude asymptotic spectral correctness of spectral Galerkin finite element
approximations of Maxwell eigenvalue problems.
