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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 3.55 on page 110 ... ...

Theorem 3.55 and its proof read as follows:

View attachment 9165

At the start of the second paragraph of the above proof by Stromberg we read the following:

" ... ...Since \(\displaystyle A_1^{ - \ \circ } = \emptyset\), we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) and then we can choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) [ check that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x) \) ] ... ...

My questions are as follows:

Can someone explain and demonstrate why/how it is that \(\displaystyle A_1^{ - \ \circ } = \emptyset\) means that we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ... how are we (rigorously) sure this is true ... ?

How/why can we choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) ... ?

... and why are we checking that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x)\) ... ... ?

*** EDIT ***

My thoughts on Question 2 ...

Since \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) is open ... \(\displaystyle \exists \ r_1\) such that \(\displaystyle B_{ r_1 } ( x_1 ) \subset V\) \ \(\displaystyle A_1^{ - }\) ...

... BUT ... how do we formally and rigorously show that ...

... we can choose an \(\displaystyle r_1\) such that the closure of \(\displaystyle B_{ r_1 } ( x_1 )\) is a subset of \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ...

... (intuitively I think we just choose \(\displaystyle r_1\) somewhat smaller yet ...)

... and further why is Stromberg talking about \(\displaystyle r_1\) between \(\displaystyle 0\) and \(\displaystyle 1\) ...?

Help will be much appreciated ...

Peter

==================================================================================

The definitions of nowhere dense, first and second category and residual are relevant ... so I am providing Stromberg's definitions ... as follows:

View attachment 9166

Stromberg's terminology and notation associated with the basic notions of topological spaces are relevant to the above post ... so I am providing the text of the same ... as follows:

View attachment 9167

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 3.55 on page 110 ... ...

Theorem 3.55 and its proof read as follows:

View attachment 9165

At the start of the second paragraph of the above proof by Stromberg we read the following:

" ... ...Since \(\displaystyle A_1^{ - \ \circ } = \emptyset\), we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) and then we can choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) [ check that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x) \) ] ... ...

My questions are as follows:

**Question 1**Can someone explain and demonstrate why/how it is that \(\displaystyle A_1^{ - \ \circ } = \emptyset\) means that we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ... how are we (rigorously) sure this is true ... ?

*Question 2*How/why can we choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) ... ?

... and why are we checking that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x)\) ... ... ?

*** EDIT ***

My thoughts on Question 2 ...

Since \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) is open ... \(\displaystyle \exists \ r_1\) such that \(\displaystyle B_{ r_1 } ( x_1 ) \subset V\) \ \(\displaystyle A_1^{ - }\) ...

... BUT ... how do we formally and rigorously show that ...

... we can choose an \(\displaystyle r_1\) such that the closure of \(\displaystyle B_{ r_1 } ( x_1 )\) is a subset of \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ...

... (intuitively I think we just choose \(\displaystyle r_1\) somewhat smaller yet ...)

... and further why is Stromberg talking about \(\displaystyle r_1\) between \(\displaystyle 0\) and \(\displaystyle 1\) ...?

Help will be much appreciated ...

Peter

==================================================================================

The definitions of nowhere dense, first and second category and residual are relevant ... so I am providing Stromberg's definitions ... as follows:

View attachment 9166

Stromberg's terminology and notation associated with the basic notions of topological spaces are relevant to the above post ... so I am providing the text of the same ... as follows:

View attachment 9167

Hope that helps ...

Peter

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