So I have made a theorem about the entropy of monotone increasing bounded functions (the proof is for my thesis). The theorem states that the entropy of the class generated by these functions is finite. I couldn't find any false statement in it as could my supervisor, but then I saw an exercise in the book "Empirical processes in M-estimators" which contradicts my theorem. The exercise states: "Verify that the class G = {g : R --> [0, 1], g increasing} is not totally bounded for the supremum norm on R". Besides that, I discovered this: Show that the space of increasing, bounded function is not totally bounded w.r.t. $\sup$-norm. Therefore my theorem has to be wrong. I would like to know what is wrong about my theorem and if I could change that with a particular condition.
I had my proof made in latex, so I could only make a screenshot. My proof of the theorem
Thanks for your help