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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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:




Browder - 1 - Theorem 8.15 ... PART 1 ... ....png
Browder - 2 - Theorem 8.15 ... PART 2  ... .....png




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:



Browder - 1 - Proposition 8.13... PART 1 ........png
Browder - 2 - Proposition 8.13... PART 2 ... ....png



Hope that helps ...

Peter
 
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