Suppose that $(u_n)_{n \in \mathbb{N}}$ and $(v_n)_{n \in \mathbb{N}}$ are bounded, and that $\lim u_n - v_n = 0$, show that they have same limit points.
So I did this:
We have $\forall \epsilon > 0, \exists N_1 \in \mathbb{N}$, such that $\forall n \geq N_1$, $|u_n - v_n| \leq \epsilon$
Let now $l \in \mathbb{R}$ be a limit point of $(u_n)$, then $\exists (\phi_n)_{n \in \mathbb{N}}$ such that $\forall \epsilon > 0, \exists N_2 \in \mathbb{N}, \forall n > N_2, |u_{\phi _n} - l| < \epsilon$
Now, let $\epsilon > 0$, and let $N = \max (N_1, N_2)$ then $\forall n > N$, $|v_{\phi_n} - l| = |v_{\phi_n} -u_{\phi _n} +u_{\phi _n}- l| \leq |v_{\phi_n} - u_{\phi _n}| +|u_{\phi _n}- l| \leq 2 \epsilon $. Thus a limit point of $u_n$ is a limit point of $v_n$. By symmetrical reasoning we prove that a limit point of $v_n$ is a limit point of $u_n$. Thus $u_n$ and $v_n$ have same limit points.
What worries in my proof is that I didn't use the hypothesis that both sequence are bounded. What did I miss?