# Proper and Continuous Mappings in R^n .... ....

• MHB
• Math Amateur
In summary, Peter is asking for help proving that a given sequence is convergent, and is only successful if the set in question is the empty set.
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MHB
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Theorem 1.8.6 ... ...

Duistermaat and Kolk"s Theorem 1.8.6 and the preceding definition regarding proper mappings read as follows:View attachment 7731In the above proof we read the following:

" ... ... Indeed let $$\displaystyle ( x_k )_{ k \in \mathbb{N} }$$ be a sequence of points in $$\displaystyle F$$ with the property that $$\displaystyle ( f( x_k ) )_{ k \in \mathbb{N} }$$ is convergent with limit $$\displaystyle b \in \mathbb{R}^p$$. ... ... "My question is as follows:

How do we be sure that such a sequence exists in $$\displaystyle F$$?Help will be appreciated ... ...

Peter

Hi, Peter.

Peter said:
How do we be sure that such a sequence exists in $$\displaystyle F$$?

You are trying to prove that $f(F)$ is closed, which means you must show that $f(F)$ contains all of its limit points. This means you start with a sequence of points in $f(F)$ and assume it is convergent. The goal is then to show that the limit is in $f(F)$. Note the language of Lemma 1.2.12(iii) says "..sequence...that is convergent to a limit..."

Furthermore, the only subset of $\mathbb{R}^{n}$ that contains no convergent sequences is the empty set. If $S$ is a non-empty set with $p\in S$, then $x_{n}=p$ for all $n$ is a convergent sequence in $S$. So the only set that fails to admit the property of which you ask is the empty set.

GJA said:
Hi, Peter.
You are trying to prove that $f(F)$ is closed, which means you must show that $f(F)$ contains all of its limit points. This means you start with a sequence of points in $f(F)$ and assume it is convergent. The goal is then to show that the limit is in $f(F)$. Note the language of Lemma 1.2.12(iii) says "..sequence...that is convergent to a limit..."

Furthermore, the only subset of $\mathbb{R}^{n}$ that contains no convergent sequences is the empty set. If $S$ is a non-empty set with $p\in S$, then $x_{n}=p$ for all $n$ is a convergent sequence in $S$. So the only set that fails to admit the property of which you ask is the empty set.
Thanks for the clear explanation ...

Appreciate your guidance and help ...

Thanks again ...

Peter

## What is a proper mapping in R^n?

A proper mapping in R^n is a function that maps every point in the domain to a unique point in the codomain. This means that for every input in the domain, there is only one output in the codomain. Additionally, the mapping must be continuous, meaning that small changes in the input result in small changes in the output.

## What is a continuous mapping in R^n?

A continuous mapping in R^n is a function that preserves the topological structure of the domain and codomain. This means that small changes in the input result in small changes in the output, and points that are close together in the domain are mapped to points that are close together in the codomain.

## What is the difference between proper and continuous mappings in R^n?

The main difference between proper and continuous mappings in R^n is that proper mappings guarantee a unique output for every input, while continuous mappings only guarantee small changes in the output for small changes in the input. Additionally, proper mappings are a stronger condition than continuous mappings, as a proper mapping must also be continuous.

## What are some common examples of proper and continuous mappings in R^n?

Examples of proper mappings in R^n include the identity mapping, where the output is equal to the input, and the constant mapping, where the output is the same for all inputs. Examples of continuous mappings in R^n include polynomial functions, trigonometric functions, and exponential functions.

## How are proper and continuous mappings used in practical applications?

Proper and continuous mappings are used in many areas of science, engineering, and mathematics. They are essential for solving optimization problems, studying the behavior of complex systems, and understanding the properties of different objects and spaces. These mappings also play a crucial role in computer graphics, medical imaging, and data analysis.

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