Limit of sequence of sets

I am trying to prove a proposition, but it's proving harder than I expected. I was wondering if someone could lead me in the right direction. Please don't give full answers. I'm just looking for a hint. The problem says:

If is a sequence of sets and , show that existis if and only if .

Thank you.

Re: Limit of sequence of sets

I don't know exactly what "the limit of a sequence of sets" has been defined to mean in your problem. However, I can show you something that I think might prove this claim for your whatever your definition there is. I'll just state what I can prove, and then leave it to you to see if that, when considered in light of whatever definition you have of "convergence sequences of sets", proves this claim. I suspect this will do the job.

(And it appears my suspicion proved wrong. Typical. (Doh) )

Given a set and , define , and then inductively .

Let .

**Claim #1:** If and , then is odd.

**Proof:** Will be by induction.

Note that .

Now let be the proposition " is odd)".

Since , have that . But also , so it follows that .

Thus have proven that , and . That proves that is true.

__Assume__ is true for some .

Let . Then is even, and . Thus true implies that .

So , but also . Thus .

Now , but also . Thus .

Have shown that true implies that and also that .

But those three statements ( true, , and ) together exactly comprise the statement .

Thus . Also have shown that is true.

Thus by induction, have proven is true . That proves Claim #1.

**Claim #2:** If and there exists such that (always here ),

then ,

and .

**Proof:**

If , then since , have that .

Next, since and , can conclude that .

By the obvious induction, .

Conversely, if , then .

The induction is clear: for will be in neither (by the definition of ), nor (by induction), and therefore won't be in .

That proves Claim #2.

Re: Limit of sequence of sets

Quote:

Originally Posted by

**johnsomeone** I don't know exactly what "the limit of a sequence of sets" has been defined to mean in your problem. However, I can show you something that I think might prove this claim for your whatever your definition there is. I'll just state what I can prove, and then leave it to you to see if that, when considered in light of whatever definition you have of "convergence sequences of sets", proves this claim. I suspect this will do the job.

Given a set

and

, define

, and then inductively

.

Let

.

**Claim #1:** If

and

, then

is odd.

**Proof:** Will be by induction.

Note that

.

Now let

be the proposition "

is odd)".

Since

, have that

. But also

, so it follows that

.

Thus have proven that

, and

. That proves that

is true.

__Assume__ is true for some

.

Let

. Then

is even, and

. Thus

true implies that

.

So

, but also

. Thus

.

Now

, but also

. Thus

.

Have shown that

true implies that

and also that

.

But theose three statements together exactly comprise the statement

.

Thus

. Also have shown that

is true.

Thus by induction, have proven

is true

. That proves the claim #1.

**Claim #2:** If

,

then

, and

.

**Proof:**
If

, then

.

Then

and

again together imply that

.

By the obvious induction,

.

Conversely, if

, then

.

The induction is clear: for

will be in neither

(by the definition of

), nor

(by induction), and therefore

won't be in

.

That proves Claim #2.

Thanks for the response. I actually found a way of proving it using the characteristic function. It is quite a short proof.