This question relates to this paper on Trust Region Policy Optimization.
I understand the reasoning behind the proof of Theorem 1 presented in Appendix A, but fail to see how the absolute value was removed in going from equation (45) to Theorem 1:
$$ |\eta(\tilde{\pi}) - L_{\pi}(\tilde{\pi})| \le \frac{4\alpha^2 \gamma \epsilon}{(1 - \gamma)^2} \tag{45}$$
$$ \eta(\tilde{\pi}) \ge L_{\pi}(\tilde{\pi}) - \frac{4\alpha^2 \gamma \epsilon}{(1 - \gamma)^2} \tag{Theorem 1}$$
This suggests that the approximation is an overestimate ($\eta(\tilde{\pi}) \le L_{\pi}(\tilde{\pi})$), but I don't see why that must be true.