if $s_n<t_n$, can we say $\lim_{n\rightarrow \infty} s_n < \lim_{n\rightarrow \infty} t_n$?

No, it's not true. Let $s_n = 1 $ and $t_n = 1 + 1/n$, for all $n\in \Bbb N$. Then $s_n < t_n$ for every $n$, but $\lim_{n\to \infty} s_n = 1 = \lim_{n\to \infty} t_n$.