Proof convergence sequence of sets

**hiddy** · February 26th 2010, 06:24 AM

hey

i've got some troubles with this one:

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

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

**Plato** · February 26th 2010, 06:38 AM

Originally Posted by hiddy

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

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 $<br /> A_n = \left( {0,\frac{1}<br /> {n}} \right],\,n \in \mathbb{Z}^ +$ .
In what sense are you saying that sequence converges?
It has an empty intersection.

**hiddy** · February 26th 2010, 07:11 AM

Originally Posted by Plato

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

Sorry i forgot to mention that: $\forall n \in \mathbb{N}$

**Plato** · February 26th 2010, 07:14 AM

Originally Posted by hiddy

Sorry i forgot to mention that: $\forall n \in \mathbb{N}$

That changes nothing.
Please answer my first question.

**hiddy** · February 26th 2010, 07:27 AM

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?

**Plato** · February 26th 2010, 07:43 AM

This what you first posted.

Originally Posted by hiddy

A sequence of sets is i monotone if $A_n \subseteq A_{n+1}$ respectively $A_n \supseteq A_{n+1}$ .
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: $A_n = \left( {1 - \frac{1}{n},1} \right),\,n \in \mathbb{Z}^ + = \mathbb{N}$
Does that sequence of sets converge?
They are nested. They are bounded.
But in what sense can on say that they converge?

**hiddy** · February 26th 2010, 08:39 AM

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.

**Plato** · February 26th 2010, 08:49 AM

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?

**hiddy** · February 26th 2010, 09:06 AM

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.

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