Yeah, it is subtle, but it is not a mistake. To convince you in another way, solving homework problem 5.4 (Bragg-Williams approximation) gives you exactly the same result as we got in lecture today. Had you put in another factor of 1/2 they would not agree.
The definition of H_i is not such that the sum on i of H_i will give back all of H. What you should imagine is that all of the spins outside of i are frozen as constants with value
Does that make sense?
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