Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\lim \sup{s_n,t_n}$ $\leq$ $(\lim \sup{s_n})(\lim \sup{t_n})$?
Can someone help explain this proof to me?
2026-03-30 16:07:45.1774886865
Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\lim \sup{s_n,t_n}$ $\leq$ $(\lim \sup{s_n})(\lim \sup{t_n})$?
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Hint:
For any $\epsilon > 0$ there is $N_s,N_t \in \mathbb{N}$ such that $s_n < \limsup s_n + \epsilon$ for all $n \geqslant N_s$ and $t_n < \limsup t_n + \epsilon$ for all $n \geqslant N_t$. Then what can you say about $s_nt_n$?