upper and lower limits of sequences

Question

Suppose that $t_n\leq s_n$ for all $n\geq N_0$, and $\{s_n\}$ converges to s. Prove that lim sup $t_n\leq s$.

I want to somehow use the fact that lim inf $t_n\leq$ lim inf $s_n$ and lim sup $t_n \leq$ lim inf $s_n$ but I don't know what to do from there.

Answer 1

Hint: Since $s_n$ converges, $\limsup s_n$ and $\liminf s_n$ are equal. Thus you get your results by taking $\liminf$ or $\limsup$ on both sides.

Answer 2

Taking the limsup both sides of inequality $t_n\leq s_n$ and using $$ \limsup s_n=\lim s_n=s $$ since $\{s_n\}$ converges to $s$.