Show that if $\lim\sup a_n = -\infty$, then $a_n\to -\infty$, and if $\lim\sup a_n = \infty$ there exists a subsequence of $a_n$ that $\to\infty$. What if $\lim\inf a_n = \pm\infty$? We're working with the extended real number system, $\bar{\mathbb{R}} = \mathbb{R} \cup\{\infty,-\infty\}$.
My work:
$$\lim\sup a_n = \inf_{n\ge1}\sup_{m\ge n} a_m = -\infty$$
is known. This means that there exists some $n'\in\mathbb{N}$ such that $\sup_{m\ge n'}a_m = -\infty$. So, for all $m \ge n'$, we have $a_m \le -\infty$. This can happen iff $a_m = -\infty$ for all $m\ge n'$. Hence, $a_n\to -\infty$.
Is this proof alright? I'm concerned about steps where I've written $a_m=-\infty$ and such, even though we know that we are working with the extended real number system. Is this okay to do? Just some uneasiness since I haven't used $=$ with infinities before.
Now assume that $$\lim\sup a_n = \inf_{n\ge1}\sup_{m\ge n} a_m = \infty \implies \sup_{m\ge n} a_m = \infty \text{ for all }n\in\mathbb{N}$$
I can intuitively see that the least upper bound of all tail sequences of $a_n$ is $\infty$, so maybe $a_n\to\infty$ itself? I'm not sure because the problem explicitly asks to produce a subsequence of $a_n$ (or show that one exists) that goes to $-\infty$. How do we do this?
Similarly for the case when $\lim\inf a_n = \pm \infty$, what can we say? I'd guess that $\lim\inf a_n = \infty$, then $a_n\to\infty$ and if $\lim\inf a_n = -\infty$ then there is a subsequence of $a_n$ that goes to $\infty$.
Here's my definition for diverging to $+\infty$:
$$x_n\to +\infty:= \forall c\in\mathbb{R}\ \exists m\in\mathbb{N}\ \forall n > m\ (x_n > c)$$
$$x_n\to -\infty:= \forall c\in\mathbb{R}\ \exists m\in\mathbb{N}\ \forall n > m\ (x_n < c)$$
Second Attempt:
$$\lim\sup_{n\to\infty} a_n = -\infty \implies \lim_{n\to\infty} \sup\{a_k:k\ge n\} = -\infty$$ $$\implies \forall c\in\mathbb{R}\ \exists m\in\mathbb{N}\ \forall n>m\ (\sup\{a_k:k\ge n\} < c)$$ $$\implies \forall c\in\mathbb{R}\ \exists m\in\mathbb{N}\ \forall n>m\ (a_k < c, k\ge n)$$
$$\implies \forall c\in\mathbb{R}\ \exists m\in\mathbb{N}\ \forall n>m\ (a_n < c) \implies \lim_{n\to\infty} a_n = -\infty$$
Similarly, $$\liminf_{n\to\infty} a_n = \infty \implies a_n\to\infty$$
Best Answer
No, your proof is not correct. There exists in general no $n'\in\mathbb{N}$ such that $\sup_{m\ge n'}a_m = -\infty$, but there exists a subsequence $(n_k)$ so that $a_{n_k}\to-\infty(k\to\infty)$ and $\sup_{m\ge n}a_m \to -\infty (n\to\infty)$. The first comes from the fact that $\limsup a_n=c$ means your biggest cluster point is $c$ and cluster points are exactly the limits of converging subsequences. So $\limsup$ is the biggest extended real number a subsequence converges. The second can either be implied by the first or you can use $\limsup_{n\to\infty} a_n=\lim_{n\to\infty}\sup_{m\geq n} a_m$, which comes from the fact that $(\sup_{m\geq n} a_m)_{n\in\mathbb N}$ is a decreasing sequence so you can exchange $\inf$ and $\lim$ in $$\limsup_{n\to\infty} a_n=\inf_{n\in\mathbb N}\sup_{m\geq n} a_m=\lim_{n\to\infty}\sup_{m\geq n}a_m. $$
What you have to show (as boink already mentioned in the comments) is that for every $M\in\mathbb R$ (especially for big negative $M$) exists a $N\in\mathbb N$ so that for all $n\geq N$ holds $a_n<M$. So choose any $M$ and now let's find such a $N$. Your first step to consider the sequence $(\sup_{m\ge n}a_m)_{n\in\mathbb N}$ was in the right direction. As already mentioned above we have $\sup_{m\ge n}a_m \to -\infty (n\to\infty)$. This means we find a $N'$ so that for every $n\geq N'$ holds $\sup_{m\ge n}a_m <M$. This implies that for every $n\geq N'$ holds $a_n<M$. This number is the $N=N'$ we were looking for and we have finished the proof.
Now consider $$\lim\sup a_n = \inf_{n\ge1}\sup_{m\ge n} a_m = \infty \implies \sup_{m\ge n} a_m = \infty \text{ for all }n\in\mathbb{N}.$$ The $\lim\sup$ means the bigest cluster point of your sequence is $\infty$, which directly implies the statement, since you find for an arbitrary big $n$ a $m>n$ so that $a_m$ is arbitrary big. If you don't wanna handle with cluster points here, you can use the fact from above $$\infty=\limsup_{n\to\infty} a_n=\lim_{n\to\infty}\sup_{m\geq n}a_m$$ and then apply that the decreasing sequence $(\sup_{m\ge n}a_m)_{n\in\mathbb N}$ converges to $\infty$ which means it already has to be $\infty$.
Your last two guess regarding $\lim\inf a_n = \pm \infty$ are correct.
What you wrote under "Second Attempt" is comletely correct.
To give you an alternative proof to the first part, you could also use the property $$ \lim\inf a_n\leq \lim\sup a_n,\tag{1}\label{tag} $$ where equality holds if and only if the limit $\lim a_n$ exists and $$\lim\inf a_n= \lim\sup a_n= \lim a_n.$$ So if $\lim\sup a_n=-\infty$ already holds, it follows form \eqref{tag} that $\lim\inf a_n=-\infty$ and the statement follows from the 'iff'-part.