Linearity of the Differential .... Junghenn Theorem 9.2.1 ....

[Math Amateur]

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on Rn"

I need some help with an aspect of Theorem 9.2.1 ...

Theorem 9.2.1 reads as follows:

Theorem 9.2.1 refers to and relies on Theorem 9.1.10 ... ... so I am providing the same ... as follows:

At the end of the proof of Theorem 9.2.1 we read the following:

" ... ... Then

$( \alpha \mathbf{f} + \beta \mathbf{g} ) ( \mathbf{a} + \mathbf{h} ) = ( \alpha \mathbf{f} + \beta \mathbf{g} ) ( \mathbf{a} ) + ( \alpha \mathbf{ df_a } + \beta \mathbf{ dg_a } ) ( \mathbf{ h } ) + \| \mathbf{ h } \|$and$\lim_{ \mathbf{ h } \rightarrow \mathbf{ 0 } } ( \alpha \eta + \beta \mu ) ( \mathbf{ h } ) = \mathbf{ 0 }$Another application of 9.1.10 completes the proof."
I don't understand the comment: "Another application of 9.1.10 completes the proof." ... ... why do we need another application of 9.1.10 ...

Doesn't the line" $( \alpha \mathbf{f} + \beta \mathbf{g} ) ( \mathbf{a} + \mathbf{h} ) = ( \alpha \mathbf{f} + \beta \mathbf{g} ) ( \mathbf{a} ) + ( \alpha \mathbf{ df_a } + \beta \mathbf{ dg_a } ) ( \mathbf{ h } ) + \| \mathbf{ h } \|$and$\lim_{ \mathbf{ h } \rightarrow \mathbf{ 0 } } ( \alpha \eta + \beta \mu ) ( \mathbf{ h } ) = \mathbf{ 0 }$ ..."actually complete the proof?
Help will be appreciated ...

Peter*** EDIT ***

I think I see what the author meant ... he is arguing that the statement:" $( \alpha \mathbf{f} + \beta \mathbf{g} ) ( \mathbf{a} + \mathbf{h} ) = ( \alpha \mathbf{f} + \beta \mathbf{g} ) ( \mathbf{a} ) + ( \alpha \mathbf{ df_a } + \beta \mathbf{ dg_a } ) ( \mathbf{ h } ) + \| \mathbf{ h } \|$and$\lim_{ \mathbf{ h } \rightarrow \mathbf{ 0 } } ( \alpha \eta + \beta \mu ) ( \mathbf{ h } ) = \mathbf{ 0 }$BY 9.1.10 ! (THAT IS APPLYING 9.1.10 AGAIN)

proves that $\alpha \mathbf{ df_a } + \beta \mathbf{ dg_a }$ is the differential of $\alpha \mathbf{f} + \beta \mathbf{g}$ ...Hence we do apply 9.1.10 again ...

Is that right ...

(If it is correct ... then my apologies for the simple confusion ... )

 

[andrewkirk]

Yes that's correct. In the second application of 9.1.10 he is using the reverse direction of the iff in Theorem 9.1.10, arguing from the existence of the function $(\alpha\eta+\beta\mu)$ to the conclusion that $(\alpha f+\beta g)$ is differentiable.

 

[Math Amateur]

Thanks Andrew ...

That helps ...

Peter