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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Proposition 4.1.8 ...Proposition 4.1.8 and its proof read as follows:View attachment 9088In the above proof by Sohrab we read the following:

" ... ... If, to get a contradiction, we assume that \(\displaystyle \xi \notin K\) is a limit point of \(\displaystyle K\), then the open cover \(\displaystyle \{ ( - \infty, \xi - 1/n) \cup ( \xi + 1/n, \infty ) \}_{ n \in \mathbb{N} }\) has no finite subcover ... ... "

My question is as follows:

How would we demonstrate rigorously that the open cover \(\displaystyle \{ ( - \infty, \xi - 1/n) \cup ( \xi + 1/n, \infty ) \}_{ n \in \mathbb{N} }\) has no finite subcover ... ...?

Help will be appreciated ...

Peter

=======================================================================================It may help readers of the above post to have access to Sohrab's definition of a limit point ... so I am providing the relevant text ... as follows ...

View attachment 9089

Hope that helps ...

Peter

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Proposition 4.1.8 ...Proposition 4.1.8 and its proof read as follows:View attachment 9088In the above proof by Sohrab we read the following:

" ... ... If, to get a contradiction, we assume that \(\displaystyle \xi \notin K\) is a limit point of \(\displaystyle K\), then the open cover \(\displaystyle \{ ( - \infty, \xi - 1/n) \cup ( \xi + 1/n, \infty ) \}_{ n \in \mathbb{N} }\) has no finite subcover ... ... "

My question is as follows:

How would we demonstrate rigorously that the open cover \(\displaystyle \{ ( - \infty, \xi - 1/n) \cup ( \xi + 1/n, \infty ) \}_{ n \in \mathbb{N} }\) has no finite subcover ... ...?

Help will be appreciated ...

Peter

=======================================================================================It may help readers of the above post to have access to Sohrab's definition of a limit point ... so I am providing the relevant text ... as follows ...

View attachment 9089

Hope that helps ...

Peter