Is $a_n < \epsilon s_n$ for a strictly increasing sequence $(a_n)$, its partial sums $(s_n)$, and all sufficiently large $n$?

Let $a_n$ be a strictly increasing sequence so that its partial sums $s_n$ tend to infinit.I want to prove that for every positive real $\epsilon$ there is a natural number $N$ so that if $n>N$, then $a_n<$$\epsilon *$$s_n$, that is, it is impossible to make a sequence such as $a_n$ so that, for a given positive real $\epsilon$, every $a_n$ is greater than a given and constant fraction of its respective $s_n$.

I'm stuck at this proof, which stems from another related question:

Let $a_n$ be a strictly increasing sequence so that its partial sums $s_n$ tend to infinit.Prove that $\lim \frac{a_1+a_3+a_5+...}{a_2+a_4+a_6+...}=1$.Currently I'm more interested in the first question, but it arose from an attempt to prove the second.