Does there exist $N \in \mathbb N$ such that $\inf\{a_N , a_{N+1}, \cdots\}= \liminf a_n$?

Question

If $\liminf a_n=m\in \mathbb R$, does there exists $N \in \mathbb N$ such that $\inf\{a_N, a_{N+1},\cdots\}= m$?

This is an intriguing question I encounter when I am learning the limit superiors and limit inferiors.

I think we can find such an $N$. But I'm struggling in proving it.

I guess it has something to do with the limit inferior being an increasing sequence.

Can anyone share their opinions/insights on this, and share some techniques dealing with limit superiors/inferiors?

Answer 1

Take : $a_n=\frac{-1}{n}$, the limit inferior of this sequence is 0, but there is no $N\in\mathbb{N}$ such that $0=inf\{a_N,a_{N+1}.....\}$ .

Answer 2

No it is not true. Just consider the sequence $-1/n$.