Hi I've spent a long time trying to understand this chain rule proof but I just can't get it... I have attached 2 pictures: the second one is an intuitive chain rule proof that turns out to be bogus and the first is the correct proof. So I am trying to understand first of all what does the second proof actually mean, and secondly what did they do to fix the problem presented by the first invalid proof. Basically any and all help for me to gain insight into this proof would be appreciated. Don't be afraid to write too much or too little because I am very lost about what they are doing. What does the "property" that they lay out (resulting in equation 7) before they start the actual proof actually mean and how does it help their proof? Why introduce epsilon? I wish there was someone here in person to walk me through it but I guess I will have to try my best to get it over the internet. Thanks a lot!