Proof of Prop 8.14 by Andrew Browder: Lv = 0

[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 9410

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 \(\displaystyle Lv \leq 0\); replacing \(\displaystyle v\) by \(\displaystyle -v\), we also find that \(\displaystyle Lv \geq 0\), so \(\displaystyle Lv = 0\). ... ... "Now the argument for \(\displaystyle Lv \leq 0\) is as follows:\(\displaystyle L(tv) + r(tv) \leq 0\) \(\displaystyle \Longrightarrow tLv \leq - r(tv)\) \(\displaystyle \Longrightarrow Lv \leq - r(tv)/ t\)... so taking the limit as \(\displaystyle t \to 0\) we have \(\displaystyle \lim_{ t \to 0 } -r(tv)/t = 0\) ...Thus \(\displaystyle Lv \leq 0\)
------------------------------------------------------------------... and ... now put \(\displaystyle v = -v\) ... then \(\displaystyle L(t(-v)) + r(t (-v)) \leq 0\) \(\displaystyle \Longrightarrow -t Lv + r(t (-v)) \leq 0\) \(\displaystyle \Longrightarrow -t Lv \leq -r(t (-v))\) \(\displaystyle \Longrightarrow Lv \geq r(- tv)/t\)... then ... ... so taking the limit as \(\displaystyle t \to 0\) we have \(\displaystyle \lim_{ t \to 0 } r(-tv)/t = 0\) ...Thus \(\displaystyle Lv \geq 0\)... BUT ...why exactly is \(\displaystyle \lim_{ t \to 0 } r(-tv)/t = 0\) ... how do we formally and rigorously demonstrate this is the case ...i.e. true ...
Help will be much appreciated ...

Peter

 

[jvicens]

Dear Peter,

Thank you for reaching out for further help in understanding Proposition 8.14 in Andrew Browder's book. It is great that you are seeking a deeper understanding and clarification on this topic.

To answer your question, we need to look at the definition of the limit of a function. In general, the limit of a function f(x) as x approaches a point c is the value that f(x) gets closer and closer to as x gets closer and closer to c. In mathematical notation, it is written as:

\lim_{x \to c} f(x) = L

This means that for any given positive number \epsilon, there exists a positive number \delta such that if 0 < |x-c| < \delta, then |f(x) - L| < \epsilon. In simpler terms, this means that as x gets closer to c, f(x) gets closer to L.

Now, let's apply this definition to the limit as t approaches 0 of r(-tv)/t. We can rewrite this as r(-tv)/t = -r(tv)/t, since r is a linear map. Using the definition of the limit, we can say that for any given positive number \epsilon, there exists a positive number \delta such that if 0 < |t| < \delta, then |r(-tv)/t - 0| < \epsilon. This means that as t gets closer to 0, r(-tv)/t gets closer to 0, which proves that \lim_{t \to 0} r(-tv)/t = 0.

I hope this helps to clarify the reasoning behind taking the limit as t approaches 0 in the proof of Proposition 8.14. If you have any further questions, please do not hesitate to ask.