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I am reading Walter Rudin's book, Principles of Mathematical Analysis.
Currently I am studying Chapter 2:"Basic Topology".
Although I can basically follow it, I am concerned that I do not fully understand the proof of Theorem 2.41 (Heine-Borel Theorem).
Rudin, Theorem 2.41 reads as follows:View attachment 3795
View attachment 3796
In the above proof we read:
" ... It remains to be shown that (c) implies (a).
If $$E$$ is not bounded, then $$E$$ contains points $$x_n$$ with
$$| x_n | \gt n$$ $$ \ \ \ \ \ $$ $$(n = 1,2,3, ... )$$.
The set $$S$$ consisting of these points $$x_n$$ is infinite and clearly has no limit point in $$R^k$$, hence has none in $$E$$. ... ... "
I cannot see how Rudin concludes that the set $$S$$ "clearly" has no limit point in $$R^k$$ ... ...
Can someone explain exactly why this is the case ... what is the formal and rigorous argument?
PeterNOTE: I apologise to MHB members for the fact that a Mac Taskbar appears in the image above ... ... I have no idea how that happened!
Currently I am studying Chapter 2:"Basic Topology".
Although I can basically follow it, I am concerned that I do not fully understand the proof of Theorem 2.41 (Heine-Borel Theorem).
Rudin, Theorem 2.41 reads as follows:View attachment 3795
View attachment 3796
In the above proof we read:
" ... It remains to be shown that (c) implies (a).
If $$E$$ is not bounded, then $$E$$ contains points $$x_n$$ with
$$| x_n | \gt n$$ $$ \ \ \ \ \ $$ $$(n = 1,2,3, ... )$$.
The set $$S$$ consisting of these points $$x_n$$ is infinite and clearly has no limit point in $$R^k$$, hence has none in $$E$$. ... ... "
I cannot see how Rudin concludes that the set $$S$$ "clearly" has no limit point in $$R^k$$ ... ...
Can someone explain exactly why this is the case ... what is the formal and rigorous argument?
PeterNOTE: I apologise to MHB members for the fact that a Mac Taskbar appears in the image above ... ... I have no idea how that happened!
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