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I am reading Walter Rudin's book, Principles of Mathematical Analysis.

Currently I am studying Chapter 2:"Basic Topology".

I have a second issue/problem with the proof of Theorem 2.43 concerning the uncountability of perfect sets in \(\displaystyle R^k\).

Rudin, Theorem 2.43 reads as follows:View attachment 3805In the above proof we read:

"Suppose \(\displaystyle V_n\) has been constructed so that \(\displaystyle V_n \cap P\) is not empty. Since every point of \(\displaystyle P\) is a limit point of P, there is a neighbourhood \(\displaystyle V_{n+1}\) such that

(i) \(\displaystyle \overline{V}_{n+1} \subset V_n

\)

(ii) \(\displaystyle x_n \notin \overline{V}_{n+1}

\)

(iii) \(\displaystyle V_{n+1} \cap P\) is not empty

... ... "I do not understand how the fact that every point of \(\displaystyle P\) is a limit point of \(\displaystyle P\) allows us to claim that there is a \(\displaystyle V_{n+1}\) such that the above 3 conditions hold ... indeed the whole thing is a bit mysterious, since there is no given process for selecting the points \(\displaystyle x_1, x_2, x_3\), ... and so \(\displaystyle x_{n+1}\) may be a considerable distance from \(\displaystyle x_n\), making it difficult for \(\displaystyle \overline{V}_{n+1} \subset V_n \) to hold ...

Can someone please explain how the above fact follows ... ...

Peter

Currently I am studying Chapter 2:"Basic Topology".

I have a second issue/problem with the proof of Theorem 2.43 concerning the uncountability of perfect sets in \(\displaystyle R^k\).

Rudin, Theorem 2.43 reads as follows:View attachment 3805In the above proof we read:

"Suppose \(\displaystyle V_n\) has been constructed so that \(\displaystyle V_n \cap P\) is not empty. Since every point of \(\displaystyle P\) is a limit point of P, there is a neighbourhood \(\displaystyle V_{n+1}\) such that

(i) \(\displaystyle \overline{V}_{n+1} \subset V_n

\)

(ii) \(\displaystyle x_n \notin \overline{V}_{n+1}

\)

(iii) \(\displaystyle V_{n+1} \cap P\) is not empty

... ... "I do not understand how the fact that every point of \(\displaystyle P\) is a limit point of \(\displaystyle P\) allows us to claim that there is a \(\displaystyle V_{n+1}\) such that the above 3 conditions hold ... indeed the whole thing is a bit mysterious, since there is no given process for selecting the points \(\displaystyle x_1, x_2, x_3\), ... and so \(\displaystyle x_{n+1}\) may be a considerable distance from \(\displaystyle x_n\), making it difficult for \(\displaystyle \overline{V}_{n+1} \subset V_n \) to hold ...

Can someone please explain how the above fact follows ... ...

Peter

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