Chain rule proof

Hello guys I have read the chain rule proof on the website and have understood it but at the bottom it shows a 'Flawed Proof' and I do not understand why that proof is Flawed.

http://math.rice.edu/~cjd/chainrule.pdf

It's something to do with limits or dividing by zero??

I expect it is because could be , which means you wouldn't be able to multiply by .

Also, I believe that the that has been used to calculate is different to the that is used to give . Since they are not the same, you shouldn't cancel them... Instead, they should have been given two different names, e.g. ...