From Understanding Machine Learning :
My three questions about the proof in the below image are:
(1.) In the red box below, why is $(b_0, b_1)$ a valid interval?
(2.) How does $b_s$ being a threshold corresponding to an ERM hypothesis $h_S$ imply that $b_s \in (b_0, b_1)$?
(3.) In the blue box, what makes this condition sufficient?
Assume $S=\{(x, y)\}_{sampled}$ is some training set. Order all the $2$-tuples by their $x$ component and just write the $y$ values to get $S=(y_{x_0}, y_{x_1}, \dots, y_{x_n}).$ As an example suppose we have $S=(1,0,1,0)$. Then we have $\large S = (1, 0_{b_1}, 1_{b_0}, 0)$. But then we have improper interval notation.
What am I misunderstanding in the proof of this lemma?
$S=(1,0,1,0)$ is not in the function class. If $h_a(x_1)=1$ and $h_a(x_2)=0$, then $x_1<x_2$. So $b_0$ is always smaller than $b_1$.