Preliminar
Though I have find some similar questions in this topic, I figured I might do one myself to build a proof of this well-known theorem for two sequences $ (x_n) \text{ , } (y_n) $:
$$\limsup (x_n + y_n) \leq \limsup(x_n) + \limsup(y_n)$$
granted that the right hand part is not of the form $\infty - \infty$. I have already proved this result for every other case but the one when one of the sequence's limit superior is $- \infty$ and the other one is a real number. My request is if someone can point me where to go now within my attempt or if it fails in any step, also some guidance on how to restart the proof from scratch with other reasoning (if it's more suitable) would be really helpful.
My attempt
Some definitions I'm allowed to use
Let $(x_n)$ be a sequence of real numbers.
We say $(x_n)$ diverges to $-\infty$ if
$$ \forall M > 0, \exists N_{M} \in \mathbb{N} \Rightarrow \forall n \geq N_{M}: \text{ } x_n < - M $$
,the limit superior of $(x_n)$ is defined by
$$ \limsup(x_n) = \inf_{N \in \mathbb{N}} \{ \sup_{n \geq N} (x_n)\} $$
and we say $(x_n)$ is bounded from above if $$ \exists M>0, \forall n \in \mathbb{N}: \text{ } x_n \leq M $$
What I did
Let $(x_n)$ and $(y_n)$ be two sequences such that $\limsup(x_n)= -\infty$ and $\limsup(y_n) = y \in \mathbb{R}$
It's not so difficult to show that under these hypothesis $(x_n)$ diverges to $-\infty$ and $(y_n)$ is bounded from above i.e.
$$ \forall K > 0, \exists n_{K} \in \mathbb{N} \Rightarrow \forall n \geq n_{K}: \text{ } x_n < - K $$
and
$$ \exists Y>0, \forall n \in \mathbb{N}: \text{ } y_n \leq Y $$
As the sum of the two limits superior is $-\infty$, I want to prove now is that $(x_n + y_n)$ diverges to $-\infty$ as that would imply its limit superior is also $-\infty$, currently it's true that given $K>0$ and for a fixed $Y>0$ from before:
$$ \exists n_{K} \in \mathbb{N}: \text{ } \forall n \geq n_{K} \Longrightarrow x_n + y_n \leq - K + Y $$
As $ -K + Y = -(K-Y) $, refering to the diverges to infinity definition I need $(K-Y)$ to be an arbitrary positive number. As $Y$ is a fixed number and $K$ is arbitrary, the arbitrary part is covered. But although I know that both $K$ and $Y$ are positive, I cannot guarantee its difference, i.e $(K-Y)$ is also positive which equals to show that $K>Y$. As I was focusing on the quantifiers of the afore definitions I realized my choice of $Y$ cannot be in general less than $K$ so I reached a dead end in this attempt of proof.
Note
I'm aware of another characterization of limit superior of a sequence (the supremum of the set of all the limits of convergent subsequences in the sequence) that could make the proof easier and I'm open to discuss answers regarding it.