Given the following two inequalities, how do I obtain the third inequality? Sorry I could not get the images being displayed. Also $$f(O)\geq d$$ I have been sitting the whole day trying to solve it but with no success. Equalities have been taken from the paper "Fast algorithms for maximizing submodular functions" by Ashwinkumar Badanidiyuru & Jan Vondrak.
1st inequality: $$\left|w_{e}\left(B_{i}, \mathrm{x}\right)-\mathrm{E}\left[f_{R\left(\mathrm{x}, B_{i}\right)}(e)\right]\right| \leq \frac{\epsilon}{r} f(O)+\epsilon \mathrm{E}\left[f_{R\left(\mathrm{x}, B_{i}\right)}(e)\right]$$ 2nd inequality: $$w_{b_{i}}\left(B_{i-1}, \mathbf{x}\right) \geq(1-\epsilon) w_{o_{i}}\left(B_{i-1}, \mathbf{x}\right)-\frac{\epsilon}{r} d$$ 3rd inequality: $$\mathbf{E}\left[f_{R\left(\mathbf{x}, B_{i-1}\right)}\left(b_{i}\right)\right] \geq(1-\epsilon) \mathbf{E}\left[f_{R\left(\mathbf{x}, B_{i-1}\right)}\left(o_{i}\right)\right]-\frac{2 \epsilon}{r} f(O)$$
I don't think that you need any properties inherent to the functions. It is rather an estimation of the inequality that I'm unable to see.