Help Understanding Andrew Browder's Proof of Proposition 8.14 from Math Analysis

[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 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

\(\displaystyle \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... But why exactly (formally and rigorously) is this the case ... ... ?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 ... ...

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 \(\displaystyle Lv \leq 0 \)... ==============================================================================

 

[Opalg]

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

\(\displaystyle \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... But why exactly (formally and rigorously) is this the case ... ... ?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 ...?

This is another case where you have to distinguish between fixed and variable vectors. Here, $v$ is fixed, but $tv$ varies, and goes to $0$ as $t\to0$. So \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t|v|} = 0\), and then you can multiply by the fixed nonzero scalar $|v|$ to get \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t} = 0\).

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 \(\displaystyle Lv \leq 0 \)...

Absolutely correct! Browder has omitted a minus sign. But his conclusion is correct.

 

[Math Amateur]

This is another case where you have to distinguish between fixed and variable vectors. Here, $v$ is fixed, but $tv$ varies, and goes to $0$ as $t\to0$. So \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t|v|} = 0\), and then you can multiply by the fixed nonzero scalar $|v|$ to get \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t} = 0\).Absolutely correct! Browder has omitted a minus sign. But his conclusion is correct.

HI Opalg ...

Your post was most helpful to me ...

Peter