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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 1.8.4 ... ...
Duistermaat and Kolk"s Theorem 1.8.4 and its proof read as follows:https://www.physicsforums.com/attachments/7716
View attachment 7717
In the above proof we read the following:
Assume $$K$$ is not bounded, Then we can find a sequence $$( x_k )_{ k \in \mathbb{N} }$$ satisfying $$x_k \in K$$ and $$\mid \mid x_k \mid \mid \ge k$$, for $$k \in \mathbb{N}$$. Obviously in this case the extraction of a convergent subsequence is impossible ... ... ... "
Question 1
Assuming Apostol's definition of a bounded set (D&K don't give one!) [see below for Apostol's definition] ... ... how do we logically and rigorously negate the definition of bounded set (since $$K$$ NOT bounded) and arrive at D&K's statement that then we can find a sequence $$( x_k )_{ k \in \mathbb{N} }$$ satisfying $$x_k \in $$K and $$\mid \mid x_k \mid \mid \ge k$$, for $$k \in \mathbb{N}$$ ... ... ?Question 2
How do we formally and rigorously demonstrate that the statement " ... we can find a sequence $$( x_k )_{ k \in \mathbb{N} }$$ satisfying $$x_k \in K$$ and $$\mid \mid x_k \mid \mid \ \ge k$$, for $$k \in \mathbb{N}$$ ... " leads to the statement ... " ... the extraction of a convergent subsequence is impossible ... ... ... " ... ... (note that although this seems plausible the rigorous demonstration that it is the case eludes me) ... ...Peter
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NOTE 1
Apostol's definition of a bounded set reads as follows:View attachment 7718NOTE 2D&K's definition of compactness and their development and comments regarding compactness may be helpful to MHB members reading the above post ... ... so I am providing the same ... as follows:
View attachment 7719
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of the proof of Theorem 1.8.4 ... ...
Duistermaat and Kolk"s Theorem 1.8.4 and its proof read as follows:https://www.physicsforums.com/attachments/7716
View attachment 7717
In the above proof we read the following:
Assume $$K$$ is not bounded, Then we can find a sequence $$( x_k )_{ k \in \mathbb{N} }$$ satisfying $$x_k \in K$$ and $$\mid \mid x_k \mid \mid \ge k$$, for $$k \in \mathbb{N}$$. Obviously in this case the extraction of a convergent subsequence is impossible ... ... ... "
Question 1
Assuming Apostol's definition of a bounded set (D&K don't give one!) [see below for Apostol's definition] ... ... how do we logically and rigorously negate the definition of bounded set (since $$K$$ NOT bounded) and arrive at D&K's statement that then we can find a sequence $$( x_k )_{ k \in \mathbb{N} }$$ satisfying $$x_k \in $$K and $$\mid \mid x_k \mid \mid \ge k$$, for $$k \in \mathbb{N}$$ ... ... ?Question 2
How do we formally and rigorously demonstrate that the statement " ... we can find a sequence $$( x_k )_{ k \in \mathbb{N} }$$ satisfying $$x_k \in K$$ and $$\mid \mid x_k \mid \mid \ \ge k$$, for $$k \in \mathbb{N}$$ ... " leads to the statement ... " ... the extraction of a convergent subsequence is impossible ... ... ... " ... ... (note that although this seems plausible the rigorous demonstration that it is the case eludes me) ... ...Peter
=========================================================================================
NOTE 1
Apostol's definition of a bounded set reads as follows:View attachment 7718NOTE 2D&K's definition of compactness and their development and comments regarding compactness may be helpful to MHB members reading the above post ... ... so I am providing the same ... as follows:
View attachment 7719