I was working on the proof that for bounded sequences $a_n$ and $b_n$, $\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty} (b_n) \leq \liminf_{n\to\infty} (a_n+b_n)$. I got to a point where I concluded that if $\lim_{n\to\infty}(\inf (\{a_k +b_k | k \geq n\}))=\lim_{n\to\infty} (\inf(\{a_k | k\geq n\})) + \lim_{n\to\infty} (\inf(\{b_k |k \geq n\}))$ then $\liminf_{n\to\infty} (a_n)+\liminf_{n\to\infty} (b_n) = \liminf_{n\to\infty} (a_n+b_n)$.
I know that this not correct and a strict inequality is possible (for example), but I'm not sure why exactly the algebraic limit theorem fails to apply here. I know the algebraic limit theorem fails for divergent sequences, but found that $\lim \inf$ and $\lim \sup$ always exist and are finite for bounded sets.
So my questions is: why does the algebraic limit theorem not apply here?