Let sequences $a_n \geq b_k$ where $b_n \to 0$.
Does $\liminf a_n \geq 0$?
Definitely $a_n$ need not tend to $0$. I don't know if its limsup or liminf can be greater than to zero though.
2026-03-27 04:38:13.1774586293
If $a_n \geq b_n$ and $b_n \to 0$, does $\liminf a_n \geq 0$?
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Hint: $$\lim_{n\to\infty}b_n = 0 \Rightarrow \liminf b_n = 0.$$