I came across this exercise : For all sequence $(a_n)_ {n\in\mathbb{N}}$ of element in $\mathbb{R}$, establish the following inequality: $$\lim_{n \to +\infty} \inf a_n \leq \lim_{n \to +\infty} \sup a_n$$
This is how is the resolution begins : Let $\alpha$ be a reel number such that : $$ \alpha < \lim_{n \to +\infty} \inf a_n = \sup_{m}[\inf_{n\geq m}a_n] $$
Are we okay to say that this $\alpha$ is not always existing, e.g. take the sequence $b_n$ defined by : $$ b_n = \begin{cases} 2& \text{ if n is even}\\ (-n)& \text{ if n is odd} \end{cases}$$ here the lim inf is equal to - infinity, so there's no way such an $\alpha$ exists right?
Thanks in advance.