What is this inequality referred to as? (related to different solutions of the system)

Question

What is this inequality referred to as? $$\overline{\lim\limits_{t\rightarrow\infty}}\varphi\left ( t \right )- \overline{\lim\limits_{t\rightarrow\infty}}\psi\left ( t \right )\leq\overline{\lim\limits_{t\rightarrow\infty}}\left [ \varphi\left ( t \right )- \psi\left ( t \right ) \right ]\leq\overline{\lim\limits_{t\rightarrow\infty}}\psi\left ( t \right )- \overline{\lim\limits_{t\rightarrow\infty}}\varphi\left ( t \right )$$ where $\varphi\left ( t \right )$ and $\psi\left ( t \right )$ are defined and bounded in $\left ( t_{0}, \infty \right )$.
I believe this answer can provide a solution (https://math.stackexchange.com/a/4128412/1190467). While I grasp some properties related to this type of problem, there are others that I am completely unfamiliar with.

Answer 1

I believe the inequality is referred to as FALSE, e.g., if $\varphi(t)=5$ and $\psi(t)=4$, then $\lim(\varphi-\psi)=\lim1=1$, while $\lim\psi-\lim\varphi=4-5=-1$, and it is not true that $1\le-1$.