Let $a_n \geq b_n$ where $b_n \leq |c_n|$ with $c_n \to 0$.
Does this imply that $$\liminf a_n \geq 0?$$
These are all real-valued sequences. I don't think it is enough to conclude.
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Of course not. Take the constant sequences $a_n = b_n = -1$ and $c_n = 0$ for all $n$.
No. As a simple counterexample, you can take $a_n = -1$, $b_n = -2$ and $c_n = 0$, all constant sequences.