Problem 2.14(a + c)
 

I am having some difficulties working through problem 2.14

For a, I understand that the number of dichotomies that H can implement at most K times the number of hypothesis that H_1 can implement, which can implement N^(dvc + 1). I'm not sure how to relate this to the dvc(H).

For c, I can prove the first half of the min function using part a. For the second part of the min function, ( 7(dvc + K) log2 (dvcK) ), I plugged the function into the inequality in b, and simplified to get:
(dk)^(7(dvc+K)) > 2K(7(dvc + K) log(dvc K))^d

I am not sure how to proceed from here, and no simplifications I can do from here seem get closer to solving the inequality.

Re: Problem 2.14(a + c)
 

For (a), if you set the break point to k*=dvc+1 for each hypothesis, it's obvious that dvc<k*. Then the upper bound of dvc for the union hypothesis set H is Kk*, meaning dvc < Kk*=K(dvc+1). This is my solution. I'm not 100% sure though.

For (c), I have the same confusion as you. I tried to simplify that inequality for a couple of hours and got nothing closer to the result. Hope someone could shed some light...