Symmetric clipping proof of transport optimality ($u_n=\max (-n,\min (u,n)),v_n=\max (-n,\min (v,n))$)

For a nonnegative cost, simultaneous symmetric clipping preserves feasibility of a pair of Kantorovich potentials. On their equality set the clipped sums are nonnegative and increase to the cost. Their bounded integrals are fixed by the marginals, so monotone convergence theorem proves optimality without subtracting undefined infinite marginal integrals.