 Solution

In physical concentration coordinates, the two nullclines areSince $K_S<K_M$, the substrate nullcline lies above the complex nullcline for $S>0$. Starting at $(S_0,0)$, the complex rises and substrate falls; with $E_0/S_0\ll 1$, this fast segment is nearly vertical when the axes use their respective concentration scales. The trajectory then follows the complex quasi-steady manifold toward the origin. On its slow descending portion it is slightly above the complex nullcline so that $\dot{C}<0$; the leading outer curve itself sets that derivative to zero only at zeroth order.