If $\sigma,\tau,\beta>0$ then all solutions of the Lorenz equation

will enter an ellipsoid centered at $(0,0,2\tau)$ in finite time. In addition the solution will remain inside the ellipsoid once it has entered. To observe this we define a Lyapunov function

It then follows that

We then choose an ellipsoid which all the solutions will enter and remain inside. This is done by choosing a constant $C>0$ such that the ellipsoid

is strictly contained in the ellipsoid

Therefore all solution will eventually enter and remain inside the above ellipsoid since $\dot{V}<0$ when a solution is located at the exterior of the ellipsoid.