I've written the following proof to show that $\lim \sup(a_n) \geq \lim \inf(a_n)$, however I'm unsure if it's right/holds. Could I get some help poking holes in it, and potentially fixing these holes?
Define $b_n$ to be the supremum of $a_n$ and define $c_n$ to be the infimum of $a_n$.
By definition, $b_n \geq c_n$ (Unsure if this is rigorous enough), which implies $b_n - c_n \geq 0$. Therefore, by a previous proof (This proof I've done shows that if elements of a sequence are within a certain range, the limit must also fall there) $\lim\ b_n - \lim\ c_n \geq 0$ must be true. This implies $\lim \sup \geq \lim \inf$.
I've also considered making a third sequence that is the difference between the supremum and the infimum, and showing that the elements of this third sequence must fall between $[0, \infty)$, however I don't see how that proves the sup is greater than or equal to the inf.