$u_n$ bounded, $v_n$ convergent then $\lim\sup(u_n + v_n) = \lim\sup (u_n) + \lim (v_n)$

Question

If $(u_n)$ and $(v_n)$ are two sequences such that $(u_n)$ is bounded and $(v_n)$ is convergent, then prove that $$\lim\sup(u_n + v_n) = \lim\sup (u_n) + \lim (v_n)$$ I have proved that $\lim\sup(u_n +v_n) \leq \lim\sup u_n + \lim v_n$. Rest part, I can't do, please help me.

Answer 1

The case in which $\limsup u_n$ is infinite is easier; do you think you can do it yourself?

Now, suppose that $\limsup u_n$ is finite and let $(n_k)_{k\in\mathbb{N}}$ be a subsequence such that

$$\lim_{k\to\infty}u_{n_k}=\limsup u_n$$

Then $\limsup (u_n +v_n)\geq \limsup(u_{n_k}+v_{n_k})=\lim u_{n_k}+\lim v_{n_k}=\limsup u_n + \lim v_n$. Here, we have used that

$\quad (1)$: If $(v_n)$ converges to $v$, then so does every subsequence of $(v_n)$.

$\quad (2)$: If $(u_n)$ converges to $u$ and $(v_n)$ converges to $v$, then $(u_n+v_n)$ converges to $u+v$.

$\quad (3)$ : $(u_n)$ converges to $u$ if and only if $\limsup u_n=\liminf u_n=u$.