# Understand Andrew Browder's Prop 8.13: Math Analysis Introduction

• MHB
• Math Amateur
In summary, Peter is trying to understand how to rigorously and validly make the move in a proof of Proposition 8.13, but is having difficulty doing so because of the modulus signs.
Math Amateur
Gold Member
MHB
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 further help in fully understanding the proof of Proposition 8.13 ...

View attachment 9404
View attachment 9405
I think that a fully detailed proof of Proposition 8.13 reads somewhat as follows:Browder's Definition 8.9 essentially means that $$\displaystyle \text{df}_p$$ exists if $$\displaystyle \lim_{ h \to 0 } \frac{1}{| h | } (f(p + h) - f(p) - \text{df}_p h ) = 0$$
Thus ... if we take $$\displaystyle \epsilon = C - \| df_p \|$$ then we can find $$\displaystyle \delta$$ such that ...$$\displaystyle | | h | \lt \delta \Longrightarrow \frac{1}{ | h | } | (f(p + h) - f(p) - \text{df}_p h ) - 0 | \leq \epsilon$$ ... ... ... (1)so that $$\displaystyle | h | \lt \delta \Longrightarrow | (f(p + h) - f(p) - \text{df}_p h ) | \leq \epsilon | h |$$ ... ... ... (2)Now the reverse triangle inequality (Duistermaat & Kolk Lemma 1.1.7 (iv) ) states that$$\displaystyle \| x - y \| \geq | \ \| x \| - \| y \| \ |$$Using the reverse triangle inequality we have $$\displaystyle | (f(p + h) - f(p) ) - ( \text{df}_p h ) | \geq | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ |$$ ... ... ... (3)Now (2) and (3) $$\displaystyle \Longrightarrow$$$$\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|$$ $$\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|$$
Now $$\displaystyle | \text{df}_p h ) | \leq | \text{df}_p | | h ) |$$ (Is that correct? ) ... so that ...$$\displaystyle | f(p + h) - f(p) | \leq | \text{df}_p | | h ) | + \epsilon |h|$$
$$\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq ( \| \text{df}_p \| + \epsilon ) |h| + C |h|$$
Is that correct?Now ... my specific problem is how to rigorously and validly make the move $$\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|$$ $$\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|$$... ... since I have effectively ignored the modulus signs around $$\displaystyle \ | f(p + h) - f(p) | - | \text{df}_p h ) |$$ ...
... that is I have assumed that $$\displaystyle | f(p + h) - f(p) | \geq | \text{df}_p h ) |$$ ...
Can someone please explain how i deal with this issue ...

Help will be much appreciated ...

Peter

#### Attachments

• Browder - 1 - Proposition 8.13... PART 1 ........png
5.4 KB · Views: 91
• Browder - 2 - Proposition 8.13... PART 2 ... ....png
2 KB · Views: 83
Last edited:
Peter said:
Now ... my specific problem is how to rigorously and validly make the move $$\displaystyle | \ | f(p + h) - f(p) | - | \text{df}_p h ) | \ | \leq \epsilon |h|$$ $$\displaystyle \Longrightarrow | f(p + h) - f(p) | \leq | \text{df}_p h | + \epsilon |h|$$... ... since I have effectively ignored the modulus signs around $$\displaystyle \ | f(p + h) - f(p) | - | \text{df}_p h ) |$$ ...
... that is I have assumed that $$\displaystyle | f(p + h) - f(p) | \geq | \text{df}_p h ) |$$ ...
Can someone please explain how i deal with this issue ...
For real numbers $X$ and $Y$, $|X|\leqslant Y$ means $-Y\leqslant X\leqslant Y$. In particular, $|X|\leqslant Y \Longrightarrow X\leqslant Y$.

In this case, $| f(p + h) - f(p) | - | \text{df}_p h | \leqslant\bigl| \ | f(p + h) - f(p) | - | \text{df}_p h | \bigr| \leqslant | \text{df}_p h ) |$.

Opalg said:
For real numbers $X$ and $Y$, $|X|\leqslant Y$ means $-Y\leqslant X\leqslant Y$. In particular, $|X|\leqslant Y \Longrightarrow X\leqslant Y$.

In this case, $| f(p + h) - f(p) | - | \text{df}_p h | \leqslant\bigl| \ | f(p + h) - f(p) | - | \text{df}_p h | \bigr| \leqslant | \text{df}_p h ) |$.

Thanks for a most helpful Post, Opalg

Peter

## 1. What is Andrew Browder's Prop 8.13?

Andrew Browder's Prop 8.13 is a mathematical analysis introduction that focuses on the fundamental concepts and techniques of mathematical analysis. It covers topics such as limits, continuity, differentiation, and integration.

## 2. Who is Andrew Browder?

Andrew Browder was an American mathematician who specialized in functional analysis and mathematical physics. He was a professor at Princeton University and the University of California, Berkeley.

## 3. Why is Prop 8.13 important?

Prop 8.13 is important because it provides a solid foundation for understanding more advanced topics in mathematics, such as complex analysis, differential equations, and topology. It also has practical applications in fields such as physics, engineering, and economics.

## 4. What are the key concepts covered in Prop 8.13?

The key concepts covered in Prop 8.13 include limits, continuity, differentiation, integration, sequences, series, and convergence. These concepts are essential for understanding the behavior of functions and their applications.

## 5. Is Prop 8.13 suitable for beginners?

Yes, Prop 8.13 is suitable for beginners as it starts with basic concepts and gradually builds upon them. However, a strong foundation in algebra and calculus is recommended before diving into this material.

• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
4
Views
2K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
2
Views
2K
• Topology and Analysis
Replies
1
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
4
Views
2K
• Topology and Analysis
Replies
2
Views
2K
• Topology and Analysis
Replies
2
Views
2K