From the definition of diverging sequences we have that
$ \forall M>0 \ \exists N $ such that $n>N \implies s_n >M $
From the first limit, we know that such an N exists since we assumed that the limit diverges. Now if I take $N' = N^2$ for the limit I want to prove, will it suffice? If I am attacking this problem the wrong way, please suggest some other way.