Problem 2.10
I understand that for the case m(N) = 2^N, we can show that this is true. But how do we prove it when m(N) < 2^N? It seems like every single theorem is giving me an upper bound. Any hints would be super appreciated.

Re: Problem 2.10
Hint: Any dichotomy 2N points can be viewed as a dichotomy on the first N points plus a dichotomy on the second N points.
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Re: Problem 2.10
I am still not quite clear about this problem. To prove this problem is true for all N values, I think we should discuss different relationship between break point k, N and 2N. I can prove that when N < 2N < k or k is infinite, the inequality holds. Also when N < k < 2N, the in equality also holds. When k < N < 2N, do you mean mH(2N) = 2mH(N)? However I did not figure it out.:clueless:

Re: Problem 2.10
@Magdon  Can you explain us in detail.
That would help! Thanks for your time! 
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