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.