after "all the other values vary, as well as fixing" b is there on wrong place (its before f). also at the end when the sum is replaced with 1/2, 1/T is missing I think. also j instead of r and j instead of v_j...

Wow, this is so incredibly helpful!! I could not have understood the proof in the textbook for a million years if not for this essay. I hope you were the author of the textbook!