I know there are multiple posts in this site about this proof but at this point I have read them all and am still struggling with a certain point.
Let $\left\{ a\_{n}\right\} \_{n\in\mathbb{N}}$ and $\left\{ b\_{n}\right\} \_{n\in\mathbb{N}}$ be bound real sequences. Then we have $$\liminf\_{n\rightarrow\infty}\left(a\_{n}\right)+\liminf\_{n\rightarrow\infty}\left(b\_{n}\right)\leq\liminf\_{n\rightarrow\infty}\left(a\_{n}+b\_{n}\right)$$

I am trying to understand the proof that uses the fact that both sequences have a common subsequence $\left\{ n\_{k}\right\} \_{k\in\mathbb{N}}$ (Whose existence I can prove) such that both $a\_{n\_{k}}\xrightarrow[k\rightarrow\infty]{}K$ and $b\_{n\_{k}}\xrightarrow[k\rightarrow\infty]{}L$. It is clear to me that $$\liminf\_{n\rightarrow\infty}\left(a\_{n}\right)+\liminf\_{n\rightarrow\infty}\left(b\_{n}\right)\leq K+L$$ But why would it be true that $$\liminf\_{n\rightarrow\infty}\left(a\_{n}+b\_{n}\right)=K+L$$