Before revision:  This will allow us to conclude that if a trajectory converges to the boundary, then a whole family of Lyapunov functions decrease along the trajectory , a fact that will allow us to overcome the fact that the usual Lyapunov function used in this setting does not diverge to infinity as trajectories converge to the boundary , which has been the technical sticking point to a proof of the Global Attractor Conjecture over the years .
After revision:  This will allow us to conclude that if a trajectory converges to the boundary, then a whole family of Lyapunov functions decrease along the trajectory . This will allow us to overcome the fact that the usual Lyapunov function used in this setting does not diverge to infinity as trajectories converge to the boundary , which has been the technical sticking point to a proof of the Global Attractor Conjecture over the years .