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I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows:

View attachment 9134

In the above proof by Stromberg we read the following:

" ... ...Next let \(\displaystyle U = \bigcap_{ k = 1 }^n U_{ y_k }\). Then \(\displaystyle U\) is a neighbourhood of \(\displaystyle x\) and \(\displaystyle U \subset S'\) ... "

My question is as follows:

It seems plausible that \(\displaystyle U \subset S'\) ... ...

... ... BUT ... ...

... how would we demonstrate rigorously that \(\displaystyle U \subset S'\) ... ... ?

(Note that \(\displaystyle S'\) is \(\displaystyle S\) complement ...)

Help will be much appreciated ... ...

Peter=================================================================================The above post mentions Hausdorff spaces ... so I am providing access to Stromberg's definition of a Hausdorff space ... as follows:

View attachment 9135I believe it may be helpful to MHB readers to have access to some of Stromberg's terminology and notation associated with topological spaces ... so I am providig access to the same ... as follows:

View attachment 9136

Hope that helps ... ...

Peter

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows:

View attachment 9134

In the above proof by Stromberg we read the following:

" ... ...Next let \(\displaystyle U = \bigcap_{ k = 1 }^n U_{ y_k }\). Then \(\displaystyle U\) is a neighbourhood of \(\displaystyle x\) and \(\displaystyle U \subset S'\) ... "

My question is as follows:

It seems plausible that \(\displaystyle U \subset S'\) ... ...

... ... BUT ... ...

... how would we demonstrate rigorously that \(\displaystyle U \subset S'\) ... ... ?

(Note that \(\displaystyle S'\) is \(\displaystyle S\) complement ...)

Help will be much appreciated ... ...

Peter=================================================================================The above post mentions Hausdorff spaces ... so I am providing access to Stromberg's definition of a Hausdorff space ... as follows:

View attachment 9135I believe it may be helpful to MHB readers to have access to some of Stromberg's terminology and notation associated with topological spaces ... so I am providig access to the same ... as follows:

View attachment 9136

Hope that helps ... ...

Peter