Validating the Chain Rule in Calculus

Jimmy Chung
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Is this proof of the chain rule valid ?
 

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How did you get this?
Did you write this yourself?
 
It's from a textbook...I just re-typed it into a word doc...
 
Is it correct...I suspect there maybe some circular reasoning with the "error term" becoming 0 as h approaches 0.
 
It looks valid to me, but just tell us why you do not trust your book. At which part.
 
Can you explain to be the part on the bottom when we set g2=g(x+h) but E(g2) turns into E(g(x)) not E(g(x+h)) ?

If that's valid, why is the limit as h approaches 0 of E(g(x)) is 0 ?
 

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