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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 \(\displaystyle K\) is not bounded, Then we can find a sequence \(\displaystyle ( x_k )_{ k \in \mathbb{N} }\) satisfying \(\displaystyle x_k \in K\) and \(\displaystyle \mid \mid x_k \mid \mid \ge k\), for \(\displaystyle k \in \mathbb{N}\). 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 \(\displaystyle K\) NOT bounded) and arrive at D&K's statement that then we can find a sequence \(\displaystyle ( x_k )_{ k \in \mathbb{N} }\) satisfying \(\displaystyle x_k \in \)K and \(\displaystyle \mid \mid x_k \mid \mid \ge k\), for \(\displaystyle k \in \mathbb{N}\) ... ... ?

**Question 2**How do we formally and rigorously demonstrate that the statement " ... we can find a sequence \(\displaystyle ( x_k )_{ k \in \mathbb{N} }\) satisfying \(\displaystyle x_k \in K\) and \(\displaystyle \mid \mid x_k \mid \mid \ \ge k\), for \(\displaystyle k \in \mathbb{N}\) ... " 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 7718

__D&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:__

**NOTE 2**View attachment 7719