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Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3 ...

  1. Aug 13, 2017 #1
    1. The problem statement, all variables and given/known data

    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:

    ?temp_hash=ae127f8cdbb2ce1c102644934fa8ac7f.png

    2. Relevant equations

    The definitions of open and closed sets are relevant as is the definition of an \epsilon neighborhood. Sohrab defines these concepts/entities as follows:

    ?temp_hash=ae127f8cdbb2ce1c102644934fa8ac7f.png


    3. The attempt at a solution

    Reflecting in general terms, I suspect the proof that ##\mathbb{N}## and ##\mathbb{Z}## are closed is approached by looking at the complements of the sets of ##\mathbb{N}## and ##\mathbb{Z}## ... visually ##\mathbb{R}## \ ##\mathbb{N}## and ##\mathbb{R}## \ ##\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 ##\epsilon##-neighbourhoods ... can someone please help ...

    I have not made any progress regarding the set ##\{ \frac{1}{n} \ : \ n \in \mathbb{N} \}## ...

    Peter
     

    Attached Files:

  2. jcsd
  3. Aug 13, 2017 #2

    FactChecker

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    Pick any real number, r (use a variable, not a specific number), not in the set and determine ε such that 2ε is the distance to the nearest number in the set. Proceed from there.
     
  4. Aug 13, 2017 #3

    Thanks FactChecker ... I am assuming you are referring to the set ##\{ \frac{1}{n} \ : \ n \in \mathbb{N} \}## and not to the proof that ##\mathbb{N}## and ##\mathbb{Z}## are closed ... is that correct ... ?

    Peter
     
    Last edited: Aug 13, 2017
  5. Aug 13, 2017 #4

    FactChecker

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    For N and Z, you can use it to prove that the complement is open. For the 1/n set, you can use it to prove that the complement is not open. It is using the definitions of open and closed sets directly, which is what you want to do unless you have some other proven theorems to use.
     
  6. Aug 13, 2017 #5
    Hi FactChecker ... I am still perplexed ...

    Can you help further ...

    Peter
     
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