If $a_n \geq b_n$, $b_n \leq |c_n|$ with $c_n \to 0$, does this imply $\liminf a_n \geq 0?$

Question

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.

Answer 1

Of course not. Take the constant sequences $a_n = b_n = -1$ and $c_n = 0$ for all $n$.

Answer 2

No. As a simple counterexample, you can take $a_n = -1$, $b_n = -2$ and $c_n = 0$, all constant sequences.