Q.1 Assume $s_n\ge a$ for all but finitely many n, then $\lim s_n\ge a$.
I think using contradiction can prove it but i wonder if there is a more direct proof.
Q.2 suppose that there exists $N_0$ such that $s_n\le t_n$for all $n>N_0$, prove that if $\lim s_n$ and $\lim t_n$ exist, then $\lim s_n\le \lim t_n$.