Now let be the operator formula_24 acting on , where is the orthogonal projection to . Denote by the polynomial such that is compact. The subspace is internal of hyperfinite dimension. By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis for where runs from to , such that each of the corresponding -dimensional subspaces is -invariant. Denote by the projection to the subspace . For a nonzero vector of finite norm in , one can assume that is nonzero, or to fix ideas. Since is a compact operator, is infinitely close to and therefore one has also . Now let be the greatest index such that formula_25. Then the space of all standard elements infinitely close to is the desired invariant subspace.