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

[Math Amateur]

Homework Statement

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:

Homework 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:

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

 

[FactChecker]

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.

 

[Math Amateur]

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.

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

 

[FactChecker]

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

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.

 

[Math Amateur]

Hi FactChecker ... I am still perplexed ...

Can you help further ...

Peter

 

[Bashyboy]

Do you know whether an arbitrary union of open sets is open; that is, if $\{U_i\}_{i \in I}$ is some collection of open intervals, do you know whether $\bigcup_{i \in I} U_i$ is open? It shouldn't be hard to prove it from the definition you provided in your OP. If so, you should be pretty straightforward to write the complement of $\Bbb{N}$ and $\Bbb{Z}$ as the union of open intervals.

 

[Bashyboy]

As for showing the set $\{1/n ~|~ n \in \Bbb{N} \}$ is neither closed nor open, consider the numbers $0$ and $1$. Using $0$ will help you show that it isn't closed, whereas $1$ will help you show that it isn't open.