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. )
Given a set and , define , and then inductively .
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 ),
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.