Writing $Df(a)$ as a sum in the definition of the derivative of a multivariable function

Question

Here is the Theorem in Spivaks Calculus on Manifolds

He eventually begins the main part of his proof by writing the definition of the derivative of $f$ and then by a process of inequalities he shows that it equals $0$. But instead of writing the standard $\dfrac{|f(a+h)-f(a)-Df(a)h|}{|h|}$ he writes $Df(a)h$ as a sum, as is seen in the image below. Why can we make this replacement?

Answer 1

You should notice three things:

1. In the sum $f(a+h)-f(a)$ all terms are cancelling except $ f(a+h) = f(a^1+h^1, \dots, a^n+h^n)$ and $ f(a) = f(a^1, \dots, a^n)$
2. By definition of the partial derivatives you have $Df(a)h = \sum\limits_{i=1}^n h^i D_if(a)$
3. Then the mean value theorem is applied on real valued functions.