If $\underset{n\to\infty}{\lim\inf}(a_n+b_n)=\underset{n\to\infty}{\lim\inf}\ a_n+\underset{n\to\infty}{\lim\inf}\ b_n$ for any sequence $\{b_n\}$ in $\Bbb R$, does $\{a_n\}$ have to be convergent?
My attempt:
This statement is somewhat converse from the one proven here. I considered two sequences: $x_n=-a_n$ and $y_n=-b_n$. Then$$-\underset{n\to\infty}{\lim\inf}(a_n+b_n)=\underset{n\to\infty}{\lim\sup}(-a_n-b_n)=\underset{n\to\infty}{\lim\sup}(x_n+y_n)$$
Instead of the definition $$\underset{n\to\infty}{\lim\inf}\ x_n=\lim_{k\to\infty}\underset{n\ge k}{\inf}\ x_n,$$I used the fact if $\{x_n\}$ is bounded in $\Bbb R,\space \underset{n\to\infty}{\lim\inf}\ x_n$, as an accumulation point of $\{x_n\}$, is a limit of a convergent subsequence $\{x_{p_n}\}_n$ of $\{x_n\}$, i. e., $\lim\limits_{n\to\infty} x_{p_n}=\underset{n\to\infty}{\lim\inf}\ x_n$.
If the corresponding sequence $\{y_n\}$ is defined by $$y_n:=\begin{cases}\varepsilon,& n\in\{p_n\}\\0, &n\notin\{p_n\}\end{cases},$$ let $\varepsilon=\underset{n\to\infty}{\lim\sup}\ x_n-\underset{n\to\infty}{\lim\inf}\ x_n$. Then $\color{blue}{-\underset{n\to\infty}{\lim\inf}(a_n+b_n)\ge-\underset{n\to\infty}{\lim\inf}\ a_n (*)}$.
Since $x_{p_n}\overset{n\to\infty}{\to}\underset{n\to\infty}{\lim\inf}$,$$(\forall\varepsilon>0)(\exists n_\varepsilon\in\{p_n\})(n\in\{p_n\}\land n>n_\varepsilon)\implies\underbrace{(|x_n-\underset{n\to\infty}{\lim\inf}\ x_n|<\varepsilon)}_{\implies x_n<\underset{n\to\infty}{\lim\inf}\ x_n+\varepsilon}$$
For $\varepsilon=\underset{n\to\infty}{\lim\sup}\ x_n-\underset{n\to\infty}{\lim\inf}\ x_n$, we thenobtain $x_n<\underset{n\to\infty}{\lim\sup}\ x_n\tag 1$.
On the other hand, $\forall n\in\Bbb N$ sufficiently large, it holds $x_n<\underset{n\to\infty}{\lim\sup}\ x_n\tag 2+\varepsilon,$ so $(1)\land(2)\implies x_n+y_n<\underset{n\to\infty}{\lim\sup} x_n+\varepsilon$, and it should hold $\underset{n\to\infty}{\lim\sup}(x_n+y_n)\le\underset{n\to\infty}{\lim\sup} x_n$, but I'm not sure how to precisely justify it. This is equivalent to $\color{blue}{-\underset{n\to\infty}{\lim\inf}(a_n+b_n)\le-\underset{n\to\infty}{\lim\inf}\ a_n(**)}$.
Finally, $(*)\land(**)\implies\underset{n\to\infty}{\lim\inf}(a_n+b_n)=\underset{n\to\infty}{\lim\inf}\ a_n\implies\underset{n\to\infty}{\lim\sup}\ y_n=\varepsilon=0\implies\{x_n\}\text{ is convergent}$.
May I ask if my arguments are valid and how to improve my proof if it makes any sense?
Thank you in advance!