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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with a part of Exercise 2.2.4 Part (3) ... ...
Exercise 2.2.4 Part (3) reads as follows:
View attachment 7185I am unable to make a meaningful start on this problem ... can someone help me with the exercise ...
Note: Reflecting in general terms, I suspect the proof that \(\displaystyle \mathbb{N}\) and \(\displaystyle \mathbb{Z}\) are closed is approached by looking at the complement sets of \(\displaystyle \mathbb{N}\) and \(\displaystyle \mathbb{Z}\) ... visually \(\displaystyle \mathbb{R}\) \ \(\displaystyle \mathbb{N}\) and \(\displaystyle \mathbb{R}\) \ \(\displaystyle \mathbb{Z}\) and proving that these sets are open ... which intuitively they seem to be ... but I cannot see how to technically write the proof in terms of open sets and \(\displaystyle \epsilon\)-neighborhoods ... can someone please help ...
I have not made any progress regarding the set \(\displaystyle \{ \frac{1}{n} \ : \ n \in \mathbb{N} \}\) ...
Peter
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The above exercise relies on the definition of open sets and related concepts and so to provide readers with a knowledge of Sohrab's definitions and notation I am provided the following text ...View attachment 7186
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with a part of Exercise 2.2.4 Part (3) ... ...
Exercise 2.2.4 Part (3) reads as follows:
View attachment 7185I am unable to make a meaningful start on this problem ... can someone help me with the exercise ...
Note: Reflecting in general terms, I suspect the proof that \(\displaystyle \mathbb{N}\) and \(\displaystyle \mathbb{Z}\) are closed is approached by looking at the complement sets of \(\displaystyle \mathbb{N}\) and \(\displaystyle \mathbb{Z}\) ... visually \(\displaystyle \mathbb{R}\) \ \(\displaystyle \mathbb{N}\) and \(\displaystyle \mathbb{R}\) \ \(\displaystyle \mathbb{Z}\) and proving that these sets are open ... which intuitively they seem to be ... but I cannot see how to technically write the proof in terms of open sets and \(\displaystyle \epsilon\)-neighborhoods ... can someone please help ...
I have not made any progress regarding the set \(\displaystyle \{ \frac{1}{n} \ : \ n \in \mathbb{N} \}\) ...
Peter
=========================================================================================
The above exercise relies on the definition of open sets and related concepts and so to provide readers with a knowledge of Sohrab's definitions and notation I am provided the following text ...View attachment 7186