Limit proof required for special case of chain rule

[lamp23]

I am trying to figure out how to prove the equality I circled below in red. I have figured out how to prove the text in blue but don't know how to use that to prove the equality I circled in red.
Below I will post the givens I'm trying to use and my guess of how to prove it.
P.S. I understand this only proves the chain rule in the special case where \(\displaystyle \Delta u \neq 0 \). This is from Stewart's Calculus and he does mention that this is not a full proof but I'm very curious how to prove this special case anyway.

I'm not sure if I'm using the right givens below.
In their blue form it looks like I will be able to use the transitive property of implication \(\displaystyle (a \rightarrow b \wedge b \rightarrow c) \rightarrow (a \rightarrow c)\) if I can always let the \(\displaystyle \epsilon\) from the 1st given equal the \(\displaystyle \delta\) from the second line.

 

[daon2]

As I tried to explain in your last thread, the proof is not merely an implication or two. Read this (or at least the counter-example on the first page), and then you will understand:

http://www-math.mit.edu/~katrin/teach/18.100/ChainRule.pdf

 

[modus]

try this. you need to scratch the notation and locate the theorem that you actually want to prove.