Integration by Parts & Change of Variables Proof

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The discussion centers on the proofs of Integration by Parts and the Change of Variables formula, specifically addressing potential typos in a textbook. The author has rewritten the proofs but is uncertain about their accuracy and seeks feedback on possible errors. A particular point of confusion involves the term g_{i-1} in the Integration by Parts proof, especially regarding its inclusion when the partition norm approaches zero. Participants question the necessity of the integration by parts approach versus using the fundamental theorem of calculus, which was preferred in their analysis course. Overall, the thread highlights the complexities and nuances involved in understanding these integration techniques.
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I'm just curious about the proofs of Integration by Parts & the Change of Variables formula
as given in this book on page 357. I think there are a lot of typo's so I've uploaded my
rewrite of them but I am unsure of how correct my rewrites are. If someone could point
out the errors & why I made them I'd really appreciate it as I don't see how I messed up
but still feel I did, for example I'm not sure whether the g_{i-1} is a typo in
the last lines of the IBP proof, when the author takes the norm of the partition to zero.
I can't see how it's included according to my rewrite :confused:

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For integration by parts, why not just integrate the product rule for differentiation?
 
The challenge I suppose. So no errors?
 
I don't see why you just done use the fundamental theorem of calculus. That's what we did in our analysis course.
 

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