  We investigate the spectra of a family of pairs (M_i,A_i) consisting of a
complete Riemannian manifold M_i and a closed subset A_i and which converge in
the Lipschitz topology to a pair (M,A). This is used to construct manifolds of
bounded curvature, nonempty essential spectrum, infinitely many eigenvalues
below the essential spectrum and with an arbitrary number of such eigenvlues of
arbitrarily high multiplicity.
