my question is as follows:
If $\{x_n\}$ and $\{y_n\}$ are bounded sequences, then $$\text{lim sup}\{x_n + y_n\} \leq \text{lim sup}\{x_n\} + \text{lim sup} \{y_n\}$$
I know that $\text{lim sup} \{x_n\}$ is the supremum of the set of all subsequential limits of $\{x_n\}$. I want to work with this definition.
How should I proceed with this definition?
Suppose that $\limsup(x_n+y_n)>\limsup(x_n)+\limsup(y_n)$. So there exists $\varepsilon>0$ such that $\limsup(x_n+y_n)>\limsup(x_n)+\limsup(y_n)+\varepsilon$, and by definition, there exists a subsequence $(x_{\varphi(n)}+y_{\varphi(n)})_n$ of $(x_n+y_n)$ which converges to some $\ell>\limsup(x)+\limsup(y)+\varepsilon$. This implies that for $n$ large enough, $x_{\varphi(n)}+y_{\varphi(n)}>\limsup(x)+\limsup(y)+\varepsilon$. But the latter implies that for $n$ large enough, either $x_{\varphi(n)}>\limsup(x)+\frac12\varepsilon$ or $y_{\varphi(n)}>\limsup(y)+\frac12\varepsilon$ (if none of those two inequalities held, we would not have the former one).
We deduce that either there exist infinitely many integers $n$ such that $x_{\varphi(n)}>\limsup(x)+\frac12\varepsilon$, or there exist infinitely many integers $n$ such that $y_{\varphi(n)}>\limsup(y)+\frac12\varepsilon$. Without loss of generality we can suppose that $x_{\varphi(n)}>\limsup(x)+\frac12\varepsilon$ holds for infinitely many integers $n$. But that would mean that $\limsup((x_{\varphi(n)})_n)\ge\limsup(x_n)_n+\frac12\varepsilon$, which is nonsense since we obviously have $\limsup((x_{\varphi(n)})_n)\le\limsup(x_n)$ (any subsequence of $(x_{\varphi(n)})_n$ is a subsequence of $(x_n)_n$).
For each $n\in\mathbb{N}$, define $s_{n}=\sup_{k\geq n}x_{k}$ and $t_{n}=\sup_{k\geq n}y_{k}$. Let $n\in\mathbb{N}$ and $k\geq n$ be arbitrary. Observe that $x_{k}+y_{k}\leq s_{n}+t_{n}$, so $\sup_{k\geq n}(x_{k}+y{}_{k})\leq s_{n}+t_{n}$. It follows that $\lim_{n\rightarrow\infty}\sup_{k\geq n}(x_{k}+y_{k})\leq\lim_{n\rightarrow\infty}s_{n}+\lim_{n\rightarrow\infty}t_{n}$. That is, $\limsup_{n}(x_{n}+y_{n})\leq\limsup_{n}x_{n}+\limsup_{n}y_{n}$.
