Thread: Limit proof required for special case of chain rule

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 . 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 if I can always let the from the 1st given equal the from the second line.

Karl's Calculus Tutor: Alternative Proof for Product of Limits

Originally Posted by Prove It

Karl's Calculus Tutor: Alternative Proof for Product of Limits

I'm not looking for a way to prove the product of the limits is the limit of the product. I'm looking for a way to prove the two expressions I circled in red are equal, i.e.