Help Understanding Andrew Browder's Proposition 8.14

[Math Amateur]

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:
View attachment 9406

In the above proof by Browder we read the following:"... ... Let \(\displaystyle L = \text{df}_p\); then \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... .. "
My question is as follows:Can someone please formally and rigorously demonstrate how Browder's definition of differentiability (Definition 8.9) ...

... ... leads to the equation \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... ..

Help will be much appreciated ...

Peter
====================================================================NOTE:

The above post mentions Browder's Definition 8.9 ... Definition 8.9 reads as follows:View attachment 9407

Hope that helps ...

Peter

 

[Opalg]

In the above proof by Browder we read the following:"... ... Let \(\displaystyle L = \text{df}_p\); then \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... .. "
My question is as follows:Can someone please formally and rigorously demonstrate how Browder's definition of differentiability (Definition 8.9) ...

... ... leads to the equation \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... ..

If you define $r(h)$ by $r(h) = f(p + h) - f(p) - Lh$ then Definition 8.9 says that \(\displaystyle \lim_{h\to0}\frac{r(h)}h = 0\), or equivalently \(\displaystyle \lim_{h\to0}\frac{r(h)}{|h|} = 0\)

 

[Math Amateur]

If you define $r(h)$ by $r(h) = f(p + h) - f(p) - Lh$ then Definition 8.9 says that \(\displaystyle \lim_{h\to0}\frac{r(h)}h = 0\), or equivalently \(\displaystyle \lim_{h\to0}\frac{r(h)}{|h|} = 0\)

Thanks Opalg ..

I appreciate the help ...

Peter