How can I prove that the infinite union of certain closed sets is not closed?

[OhMyMarkov]

Hello everyone!

I'm trying to find a set a closed set ${A_n}$ whose inifinte union is not closed. Now, I can picture the following:

If I let $A_n=[-\frac{1}{n}, \frac{1}{n}]$, then $A_n$ is closed, but their union is not, simply because the point $x=0$ seems to be a limit point at """""infinity"""", but it is not in any of the $\{A_n\}$, so the union is therefore not closed.

Now, of course, I'm having trouble showing that, because, I can't always find an $r>0$ s.t. the neighborhood of center $0$ and radius $r$ intersects any of the $\{A_n\}$s.

Similarly, I'm having trouble proving that, if $B_n = [-1+\frac{1}{n}, 1-\frac{1}{n}]$, then their infinite union is not closed.
I would appreciate any help in this... :)

 

[Plato2]

Hello everyone!
I'm trying to find a set a closed set ${A_n}$ whose inifinte union is not closed. Now, I can picture the following:
If I let $A_n=[-\frac{1}{n}, \frac{1}{n}]$, then $A_n$ is closed, but their union is not, simply because the point $x=0$ seems to be a limit point at """""infinity"""", but it is not in any of the $\{A_n\}$, so the union is therefore not closed.

Now, of course, I'm having trouble showing that, because, I can't always find an $r>0$ s.t. the neighborhood of center $0$ and radius $r$ intersects any of the $\{A_n\}$s.

Similarly, I'm having trouble proving that, if $B_n = [-1+\frac{1}{n}, 1-\frac{1}{n}]$, then their infinite union is not closed.

You are confused on notation.
$\bigcup\limits_n {\left[ { - \frac{1}{n},\frac{1}{n}} \right]} = \left[ { - 1,1} \right]$ which is a closed set.

$\bigcup\limits_n {\left[ { - 1 + \frac{1}{n},1 - \frac{1}{n}} \right]} = \left( { - 1,1} \right)$ which is an open set.
Note that $\left( { - 1 + \frac{1}{n}} \right) \to - 1$ is a decreasing sequence.
Note that $\left( {1 - \frac{1}{n}} \right) \to 1$ is an increasing sequence.
What is your question?

 

[OhMyMarkov]

Hi Plato!

Thanks for your reply.

My question is, how can I prove that the union of $[-1+\frac{1}{n}, 1-\frac{1}{n}]$ tends to $(-1,1)$ as $n$ tends to infinity?

 

[CaptainBlack]

Hi Plato!

Thanks for your reply.

My question is, how can I prove that the union of $[-1+\frac{1}{n}, 1-\frac{1}{n}]$ tends to $(-1,1)$ as $n$ tends to infinity?

I will assume that you have no trouble showing that \(x \in (-1,1)\) is in at least one of the sets: \([-1+\frac{1}{n}, 1-\frac{1}{n}],\ \ n \in \mathbb{N}_+\) and hence in their union . Also that you have no trouble showing that \(\pm 1\) are in none of \([-1+\frac{1}{n}, 1-\frac{1}{n}], \ \ n\in \mathbb{N}_+\) and hence not in their union.

(I am also assuming you have no problem for x >1 or x<-1 either)

Then you are done.

CB

 

[blue_raver22]

Hi there,

It seems like you are on the right track with your examples. When dealing with sets that are infinite, it can be tricky to prove properties like closure. However, one approach you could take is to use the definition of a closed set, which is that it contains all of its limit points. In the case of your first example, the point 0 is a limit point of the union, but it is not contained in any of the individual sets $A_n$. Therefore, the union is not closed.

For your second example, you could use a similar approach. Show that the point 1 is a limit point of the union, but it is not contained in any of the individual sets $B_n$. This would prove that the union is not closed.

I hope this helps! Keep up the good work in exploring the concept of infinity in mathematics. It can be a challenging but fascinating topic. Let me know if you have any further questions or need clarification on anything.