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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 yet further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

View attachment 9409

In the above proof by Browder, we read the following:" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt 0\) sufficiently small, we find (taking \(\displaystyle h = tv\) above) that \(\displaystyle L(tv) + r(tv) \leq 0\), or \(\displaystyle Lv \leq r(tv)/t\), so letting \(\displaystyle t \to 0\) we have \(\displaystyle Lv \leq 0\). ... ... Now ... the above quote implies that

\(\displaystyle \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ...

Peter

==============================================================================EDIT:

Just noticed that in the above quote, Browder argues that \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq r(tv)/t\) ... ... ... BUT ... i suspect he should have written \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq - r(tv)/t\) ... ..... however ... in either case ... when we let \(\displaystyle t \to 0\) we get the same result ... namely \(\displaystyle Lv \leq 0 \)... ==============================================================================

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

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

Proposition 8.14 reads as follows:

View attachment 9409

In the above proof by Browder, we read the following:" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt 0\) sufficiently small, we find (taking \(\displaystyle h = tv\) above) that \(\displaystyle L(tv) + r(tv) \leq 0\), or \(\displaystyle Lv \leq r(tv)/t\), so letting \(\displaystyle t \to 0\) we have \(\displaystyle Lv \leq 0\). ... ... Now ... the above quote implies that

\(\displaystyle \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ...

*I note that we have that \(\displaystyle \lim_{ h \to 0 } \frac{ r(h) }{ |h| } = \lim_{ t \to 0 } \frac{ r(tv) }{ tv } = 0\)... but this is (apparently anyway) not exactly the same thing ... we need to be able to demonstrate rigorously that\(\displaystyle \lim_{ h \to 0 } \frac{ r(h) }{ |h| } = \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... ... but how do we proceed to do this ...?Hope someone can help ... ...***But why exactly (formally and rigorously) is this the case ... ... ?**Peter

==============================================================================EDIT:

Just noticed that in the above quote, Browder argues that \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq r(tv)/t\) ... ... ... BUT ... i suspect he should have written \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq - r(tv)/t\) ... ..... however ... in either case ... when we let \(\displaystyle t \to 0\) we get the same result ... namely \(\displaystyle Lv \leq 0 \)... ==============================================================================

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