Question: Let $\{u_n\}_n$ and $\{v_n\}_n$ be two bounded sequences such that $\{v_n\}_n$ is convergent. Prove that $\overline{\lim}~(u_n+v_n)=\overline{\lim}~u_n+ \lim v_n$.
My try: Since $\{u_n\}_n$ is bounded, there exists a $K>0$ such that $|u_n|\leq K$ for all $n\in \mathbb{N}$. Again since $\{v_n\}_n$ is convergent, let $\lim\limits_{n\to\infty} v_n=l$. Then given $\epsilon>0$, there exists a $m\in \mathbb{N}$ such that $$|v_n-l|<\epsilon, \forall n\geq m.$$
How can I show that required result?
