  **Lemma 0**: $P_{OptSample}(T, L) \geq {P_{ROS}(T, L)}$.  **Lemma 0**: $P_{OptSample}(T, L) \geq {P_{0-Sample}(T, L)}$.  **Lemma 0**: $min(C, \frac{L}{L + 0}) \leq \frac{P_{OptSample}(T, L)}{P_{Oracle}(T, L)} \leq 0$, with $C = 0 - \ln{0} + \ln{\ln{0}} \approx 0.00$.  