problem in deriving error of taylor polynomial

[fisher garrry]

This is a derivation for the remainder term of a taylor polynomial that I have written down from kahn academy's internet site. This is the link:

Taylor polynomial remainder (part 2) (video) | Khan Academy

The more terms we have in a Taylor polynomial approximation of a function, the closer we get to the function. But HOW close? In this video, we prove the Lagrange error bound for Taylor polynomials.

www.khanacademy.org

I don't understand how they obtain the constant C. It seems to me that they go from using

[MATH]\leq[/MATH]
to using

=

But in the derivation after they put the first C=-Ma they continue to use

[MATH]\leq[/MATH]
I have a bit of trouble quantifying this?

 

[fisher garrry]

I did after some struggle manage to come up with my own version of the proof that harmonizes with the formula other places


This formula and usage of c seems to be used in other proofs. Thanks for your time!