Is it true that $A_n \to A \iff A_{n+p} \to A$ for $p \in \mathbb{N}$, where $A_n, n \geq 0$ are sets?
I can show that "$\implies$" holds, but I'm not making progress with the other implication. Any ideas?
EDIT: $A_n \to A$ means that $\limsup A_n = A = \liminf A_n$
If $x$ belongs to infinitely many elements of $A_{n+p}$, then it belongs to infinitely many elements of $A_n$ and converselly if an element belongs to infinitely many elements of $A_n$ then it belongs to infinitely many elements of $A_{n+p}$. Therefore $\limsup_n A_n=\limsup_n A_{n+p}$.
If an element belongs to all $A_{n+p}$ for $n$ sufficiently large, $\forall n>N$ for some $N$, then so does for all $A_n$ for $n$ sufficiently large just take $n>N+p$. Conversely, if $x$ belongs to all $A_{n}$ for $n$ sufficiently large, then so does for all $A_{n+p}$ for $n$ large. Therefore $\liminf_n A_n=\liminf A_{n+p}$.
The implication when you know that one of the limits exist is a particular case.