# Proof convergence sequence of sets

• Feb 26th 2010, 06:24 AM
hiddy
Proof convergence sequence of sets
hey

i've got some troubles with this one:

A sequence of sets is i monotone if $\displaystyle A_n \subseteq A_{n+1}$ respectively $\displaystyle A_n \supseteq A_{n+1}$.

Show that every monotonic sequence of sets converged and calulate the limes.

I know that if it converges the lim sup equals the lim inf but how do i show that?

thx
• Feb 26th 2010, 06:38 AM
Plato
Quote:

Originally Posted by hiddy
A sequence of sets is i monotone if $\displaystyle A_n \subseteq A_{n+1}$ respectively $\displaystyle A_n \supseteq A_{n+1}$.

Show that every monotonic sequence of sets converged and calulate the limes.

I know that if it converges the lim sup equals the lim inf but how do i show that?

Consider this sequence $\displaystyle A_n = \left( {0,\frac{1} {n}} \right],\,n \in \mathbb{Z}^ +$.
In what sense are you saying that sequence converges?
It has an empty intersection.
• Feb 26th 2010, 07:11 AM
hiddy
Quote:

Originally Posted by Plato
Consider this sequence $\displaystyle A_n = \left( {0,\frac{1} {n}} \right],\,n \in \mathbb{Z}^ +$.
In what sense are you saying that sequence converges?
It has an empty intersection.

Sorry i forgot to mention that: $\displaystyle \forall n \in \mathbb{N}$
• Feb 26th 2010, 07:14 AM
Plato
Quote:

Originally Posted by hiddy
Sorry i forgot to mention that: $\displaystyle \forall n \in \mathbb{N}$

That changes nothing.
• Feb 26th 2010, 07:27 AM
hiddy
I still don't get what you mean, that sequence you posted gets smaller with every step and goes to 0 which is not part of the sets.

Do you mean that similar to sequences of real numbers you net a monotonic and bounded seq?
• Feb 26th 2010, 07:43 AM
Plato
This what you first posted.
Quote:

Originally Posted by hiddy
A sequence of sets is i monotone if $\displaystyle A_n \subseteq A_{n+1}$ respectively $\displaystyle A_n \supseteq A_{n+1}$.
Show that every monotonic sequence of sets converged and calulate the limes.

Now I ask you: What does it mean to say that a sequence of sets converges?
Here is another example: $\displaystyle A_n = \left( {1 - \frac{1}{n},1} \right),\,n \in \mathbb{Z}^ + = \mathbb{N}$
Does that sequence of sets converge?
They are nested. They are bounded.
But in what sense can on say that they converge?
• Feb 26th 2010, 08:39 AM
hiddy
I give up, not my day not my question

The only thing left i can add is that in the last example they converge from below but that doesnt help me much.
• Feb 26th 2010, 08:49 AM
Plato
Quote:

Originally Posted by hiddy
I give up, not my day not my question
The only thing left i can add is that in the last example they converge from below but that doesnt help me much.

Are you working with Moore-Smith convergence?
• Feb 26th 2010, 09:06 AM
hiddy
ahm actually no, because my question is from basic probability theory we did not talk about nets. I know that nets are generalisiations of sequences but there hast to be an easy proof that an isotone sequence of sets converges.