(Solved): 12.4 Show limsup(sn+tn)limsupsn+limsuptn for bounded sequences (sn) and (tn). ...

12.4 Show $\lim \sup \left(s_n+t_n\right)?\mathrm{limsup}{s}_n+\lim \sup t_n$ for bounded sequences $\left(s_n\right)$ and $\left(t_n\right)$. Hint: First show $\sup \left\{s_n+t_n:n>N\right\}?\sup \left\{s_n:n>N\right\}+\sup \left\{t_n:n>N\right\}.$ Then apply Exercise 9.9(c). Screenshot 2023-07-09 at 19.21... (i) $?\stackrel{?}{?}$ 9.9 Suppose there exists $N_0$ such that $s_n?t_n$ for all $n>N_0$. (a) Prove that if $\lim s_n=+?$, then $\lim t_n=+?$. (b) Prove that if $\lim t_n=??$, then $\lim s_n=??$. (c) Prove that if $\lim s_n$ and $\lim t_n$ exist, then $\lim s_n?\lim t_n$.