Let $s_n$ and $t_n$ be bounded sequences of nonegative numbers. Prove $\text{lim sup} (s_nt_n) \leq (\text{lim sup} s_n)(\text{lim sup}$ t_n).
EDIT: new approach. Since $s_n \geq 0$ and $t_n \geq0$ then $0\leq\{s_k:k\geq n\}$ and $0\leq\{t_k:k\geq n\}$.
Now suppose $\epsilon >0$ and for $n\in \mathbb N$ and $k \geq n$ that $s_kt_k \leq (s_k) \text{sup} \{t_k:k\geq n\} \leq \text {sup} \{s_k:k\geq n\}\text{sup}\{t_k:k\geq n\}$ where I used that multiplication of nonegative numbers is preserved (and I followed from a different proof and am honestly sort of lost on this).
Thus $\text{sup}\{s_kt_k:k\geq n\} \leq \text{sup}\{s_k:k\geq n\}\text{sup}\{t_k:k\geq n\}$ which would follow for limits so $\text{lim sup} (s_nt_n) \leq (\text{lim sup}s_n)(\text{lim sup} t_n)$.