# The Chain Rule in n Dimensions ... Browder Theorem 8.15

#### Peter

##### Well-known member
MHB Site Helper
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 help in order to fully understand the proof of Theorem 8.15 ...

Theorem 8.15 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... Then $$\displaystyle |k| \leq C |h|$$ for $$\displaystyle |h|$$ sufficiently small, if $$\displaystyle C \gt \| T \|$$, by Proposition 8.13; it follows that

$$\displaystyle \frac{ r_g ( k(h) ) }{ |h| } \to 0$$ as $$\displaystyle h \to 0$$. ... ... "

My question is as follows:

Can someone demonstrate formally and rigorously how/why

$$\displaystyle \frac{ r_g ( k(h) ) }{ |h| } \to 0$$ as $$\displaystyle h \to 0$$. ... ...

Help will be much appreciated ...

Peter

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The above post mentions Proposition 8.13 ... Proposition 8.13 reads as follows:

Hope that helps ...

Peter

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