Is the following proof for $\limsup a_n+b_n\ge\liminf a_n+\limsup b_n$ correct?

Question

$a_{n} , b_{n}$ are both bounded series, so both of them have a converged subsequence. I'll define: $(a_{n_k}), (b_{n_k})$ as such: $\lim(a_{n_k}) = \liminf(a_{n}), \lim(b_{n_k}) = \limsup(b_{n})$

Now I can can claim according to the definition of the limsup: $\limsup (a_{n}+b_{n})\ge \lim(a_{n_k} + b_{n_k}) = \lim(a_{n_k}) + \lim(b_{n_k}) = \liminf(a_n) + \limsup(b_{n})$

Answer 1

Note that \begin{equation*} \liminf a_n = -\limsup -a_n. \end{equation*} Then use the fact that \begin{equation*} \limsup c_n +d_n \leq \limsup c_n +\limsup d_n \end{equation*}
Specifically, let $c_n=b_n+a_n$ and $d_n=-a_n$. Then \begin{equation*} \limsup b_n = \limsup (b_n + a_n -a_n) \leq \limsup (b_n+a_n) + \limsup -a_n. \end{equation*} Rearranging terms, we have \begin{equation*} \limsup b_n - \limsup -a_n \leq \limsup (b_n+a_n) \end{equation*} or \begin{equation*} \limsup b_n + \liminf a_n \leq \limsup (b_n+a_n). \end{equation*}

proof of claim: Let $\{c_{n_k}+d_{n_k}\}$ be a subsequence converging to $\limsup_n c_n+d_n$. Now, $\limsup_k c_{n_k} \leq \limsup_n c_n $ and $\limsup_k d_{n_k} \leq \limsup_n d_n $ since $\{n_k\} \subset \mathbb{N}$.

Answer 2

Consider $\{n_k\}$ s.t $(a_{n_k})$ goes to $\limsup (a_n)$. $\{(a+b)_{n_k}\}$ is at least $\limsup(a_n) + \liminf (b_n)$.