Nested Open Sets: Example & Intersection

[cragar]

Homework Statement

Give an example of an infinite collection of nested open sets.
$o_1 \supseteq o_2 \supseteq o_3 \supseteq o_4 ...$
Whose intersection $\bigcap_{n=1}^{ \infty} O_n$ is
closed and non empty.

Homework Equations

A set $O \subseteq \mathbb{R}$ is open if for all points, $a \in O$
there exists an $\epsilon$ neighborhood $V_{\epsilon}(a) \subseteq O$

The Attempt at a Solution

It seems like if we started with the open interval (0,1) and then took a smaller interval that was nested inside the original interval, and then just kept doing this until we enclosed one point in the interval.

 

[micromass]

It seems like if we started with the open interval (0,1) and then took a smaller interval that was nested inside the original interval, and then just kept doing this until we enclosed one point in the interval.

Yes, that's the idea.

Can you make this rigorous??

 

[cragar]

Can I just take the middle one-third of the set.
so after n operations i will have $\frac{1}{3^n}$

 

[jgens]

I cannot really tell where you are going with that last response. It might be a little easier if you consider intervals of the form $(-\frac{1}{n},\frac{1}{n})$.

 

[micromass]

Can I just take the middle one-third of the set.
so after n operations i will have $\frac{1}{3^n}$

What do you mean?? 1/3^n is just a number.

Just find sets that get smaller and smaller each time.

start with ]-1,1[, then ]-1/2,1/2[. Then what??

 

[cragar]

ok so like jgens said use $( \frac{-1}{n} , \frac{1}{n})$
And then eventually after n goes to infinity I will have 0 as my enclosed point.
so if I make an $\epsilon$ radius around 0 i will contain points inside of O the original set. Would the set zero it self be closed be cause if we make
an $\epsilon$ radius around 0 it won't contain elements that are in the set zero itself.

 

[HallsofIvy]

Or, if you really want $1/3^n$, you could use $O_n= (-1/3^n, 1/3^n)$.

 

[cragar]

ok, thanks everyone for the help