Checking the validation of a theorem

Question

I came across with a question that made me wonder if a spesfic theorem is true:

Let $a_n$ be a sequence so for every $n\in \mathbb{N}$ we get $m_1\leq a_n \leq m_2$ for some $m_1,m_2\in\mathbb{R}$. Also we know that the limit of the sequence is $L\in\mathbb{R}$. Then we get $\liminf a_n = m_1$.

If the theorem is not true, what condition should make it?

Answer 1

The theorem is not true. For example, take $a_n=0$, i.e. the constant sequence $0,0,0,0,0,0\dots$. Also, set $m_1=-9384729034809324095347985$, and $m_2=0$. Then, it is true that

For every $n\in\mathbb N$, we get $m_1\leq a\leq m_2$.

However, it is not true that $\liminf a_n=m_1$.

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In fact, if the limit of the sequence is $L$, then $\liminf a_n$ is also equal to $L$. So the only way for $\liminf a_n$ to equal $m_1$ is for $L$ to equal $m_1$.