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

I need help with an aspect of the proof of Theorem 1.6.2 ...

Duistermaat and Kolk"s Theorem 1.6.2 and its proof read as follows:View attachment 7708In the above proof we read the following:

" ... ... By the definition of supremum, only a finite number of \(\displaystyle x_k\) satisfy \(\displaystyle a + \delta \lt x_k\), while there are infinitely many \(\displaystyle x_k\) with \(\displaystyle a - \delta \lt x_k\) ... ... "Can someone please explain (very slowly and simply if you will ... :) ... ) how/why only a finite number of \(\displaystyle x_k\) satisfy \(\displaystyle a + \delta \lt x_k\), while there are infinitely many \(\displaystyle x_k\) with \(\displaystyle a - \delta \lt x_k\) ... ... Help will be much appreciated ... ...

Peter

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MHB members reading the above post may be assisted by having access to Theorem 1.6.1 and the notes succeeding the Theorem ... ... so I providing the same ... as follows ... ...https://www.physicsforums.com/attachments/7709Hope that helps ...

Peter