Given $\{a_n\}_{n=0}^\infty$ and $\{b_n\}_{n=0}^\infty$ bounded sequences; show that if $\lim \limits_{n\to \infty}a_n-b_n=0$ then both sequences have the same subsequential limits.
My attempt to prove this begins with: Let $E_A=\{L|L$ subsequential limit of $a_n$}
and $E_B=\{L|L$ subsequential limit of $b_n$}. We need to show that $E_A=E_B$.
Given bounded sequence $a_n$ and $b_n$ we know from B.W that each sequence has a subsequence that converges, therefore both $E_A$ and $E_B$ are not empty;
Let $L\in E_A$.
How can I show that $L\in E_B$?
Thank you very much.
Best Answer
You can approach it very directly. Let $L\in E_A$. Then there is a subsequence $\langle a_{n_k}:k\in\Bbb N\rangle$ converging to $L$; now use the hypothesis that $\langle a_n-b_n:n\in\Bbb N\rangle\to 0$ to show that $\langle b_{n_k}:k\in\Bbb N\rangle$ converges to $L$.
Note that you really only have to prove that $E_A\subseteq E_B$: if $\langle a_n-b_n:n\in\Bbb N\rangle\to 0$, then clearly also $\langle b_n-a_n:n\in\Bbb N\rangle\to 0$, and you can appeal to the first half to conclude that $E_B\subseteq E_A$.