Limits of Subsequences If $s=\{s_n\}$  and $t=\{t_n\}$  are two nonzero decreasing sequences converging to 0, such that $s_n ≤t_n$   for all $n$. Can we find subsequences $s ′$ of $s$  and $t ′$ of  $t$  such that $\lim \frac{s'}{t'}=0$ , i.e., $s ′$   decreases more rapidly than $t ′$  ? 

Yes, we can. So we have two positive decreasing sequences with $s_n \leq t_n$, and $s_n \to 0, t_n \to 0$.
Then we can let $t' \equiv t$. As $\{t_n\}$ is positive, $t_1 > 0$. As $s_n \to 0$, there is some $k$ s.t. $s_k < t_1/1$. Similarly, there is some $l > k$ s.t. $s_l < t_2/2$. Continuing in this fashion, we see that we can find a sequence $s'$ so that $s'_n < t_n/n$, so that $\dfrac{s'_n}{t_n} < \dfrac{1}{n}$.
