Show that every sequence in $\mathbb{R}$ has a monotone subsequence

You don't need to divide this into divergent/convergent sequences. Maybe this hint will already do:

For the sequence $(a_n)_{n\in\mathbb N}$ we call $k\in\mathbb N$ a spike of the sequence, if $a_k\geq a_n$ for all $n\geq k$. Now think about what happens if there are infinte many spikes? What if there are only finite many spikes?

if any value has infinite support we are done. otherwise there is an infinite subsequence of positive (or negative) terms $r_j (-r_j)$.

if the sequence is unbounded, we are done.

otherwise by the Heine-Borel theorem there is an accumulation point $a$. remove any terms equal to $a$ and after this reduction let $r^+$ and $r^-$ denote the subsequences of terms $\gt a$ and $\lt a$ respectively. at least one must be infinite. suppose this is $r^+$

set $k_1=1$ since $r^+_{k_1} \gt a$ we can find $r^+_{k_2} \in (a,r^+_{k_1})$

the subsequence $r^+_{k_n}$ is monotonic decreasing