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

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.

• No: $a_n=b_n=-n$ and $c_n=1/n$ – rafforaffo Jun 2 '15 at 14:03
• Perhaps you mean $|b_n| \le c_n$. – GEdgar Jun 2 '15 at 14:09
• @GEdgar Does that change the answer? – C_Al Jun 2 '15 at 15:22
• I think if so then the answer is yes, just by using relationship of liminf and limsup. – C_Al Jun 2 '15 at 15:27

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.