Does the convergence of $s_n $ imply the convergence of $t_k$, and vice versa?

Question

If $s_n $ and $t_k $ satisfy the conditions

I) $0 ≤ s_n≤ t_k $, if $n <2^k $

II) $2s_n≥ t_k≥ 0 $, if $n>2^k $

Can we conclude that $s_n $ converges if and only if $t_n $ converges?

Answer 1

For all $n\in\mathbb N$, consider $$0\leq s_n\leq t_{\lfloor \log_2 n\rfloor+1}$$ $$0\leq t_n\leq 2s_{2^n +1}$$ and if one of two sequences converges, the other does too.