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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 Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows:

View attachment 9143

In the above proof by Stromberg we read the following:

" ... ... Also, if \(\displaystyle x \in X\) and \(\displaystyle \epsilon \gt 0\), it follows from (2) that \(\displaystyle B_\epsilon (x) \cap A \supset B_{ 1/n } (x) \cap A_{ 1/n } \neq \emptyset\) , where \(\displaystyle 1/n \lt \epsilon\) ... ... "My question is as follows:

Can someone please demonstrate rigorously why/how it is the case that \(\displaystyle B_\epsilon (x) \cap A \supset B_{ 1/n } (x) \cap A_{ 1/n }\) ... ... ?

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

*** EDIT ***

After a little reflection this issue may be straightforward ... ... Wish to show formally that \(\displaystyle B_{ 1/n } (x) \cap A_{ 1/n } \subset B_\epsilon (x) \cap A \)We need to show that \(\displaystyle x \in B_{ 1/n } (x) \cap A_{ 1/n } \Longrightarrow x \in B_\epsilon (x) \cap A\)But ... leaving out details ... we have ...\(\displaystyle x \in B_{ 1/n } (x) \cap A_{ 1/n }\)\(\displaystyle \Longrightarrow x \in B_{ 1/n } (x) \text{ and } x \in A_{ 1/n }\) \(\displaystyle \Longrightarrow x \in B_{ \epsilon } (x) \text{ and } x \in A\)\(\displaystyle \Longrightarrow x \in B_\epsilon (x) \cap A\)

Is that correct?

=======================================================================================Help will be appreciated ...

Peter

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

I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows:

View attachment 9143

In the above proof by Stromberg we read the following:

" ... ... Also, if \(\displaystyle x \in X\) and \(\displaystyle \epsilon \gt 0\), it follows from (2) that \(\displaystyle B_\epsilon (x) \cap A \supset B_{ 1/n } (x) \cap A_{ 1/n } \neq \emptyset\) , where \(\displaystyle 1/n \lt \epsilon\) ... ... "My question is as follows:

Can someone please demonstrate rigorously why/how it is the case that \(\displaystyle B_\epsilon (x) \cap A \supset B_{ 1/n } (x) \cap A_{ 1/n }\) ... ... ?

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

*** EDIT ***

After a little reflection this issue may be straightforward ... ... Wish to show formally that \(\displaystyle B_{ 1/n } (x) \cap A_{ 1/n } \subset B_\epsilon (x) \cap A \)We need to show that \(\displaystyle x \in B_{ 1/n } (x) \cap A_{ 1/n } \Longrightarrow x \in B_\epsilon (x) \cap A\)But ... leaving out details ... we have ...\(\displaystyle x \in B_{ 1/n } (x) \cap A_{ 1/n }\)\(\displaystyle \Longrightarrow x \in B_{ 1/n } (x) \text{ and } x \in A_{ 1/n }\) \(\displaystyle \Longrightarrow x \in B_{ \epsilon } (x) \text{ and } x \in A\)\(\displaystyle \Longrightarrow x \in B_\epsilon (x) \cap A\)

Is that correct?

=======================================================================================Help will be appreciated ...

Peter

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