By using just the definition of a Cauchy Sequence, how could I show $s_n=n^2$ is not a Cauchy Sequence? Here's what I have,
Let $\epsilon>0$. Then there exists N such that $m,n>N \implies |n^2-m^2|<\epsilon.$
So now we have to find a $N$ that makes the implication true. So, $|n^2-m^2|<\epsilon=n^2<\epsilon+m^2=n<\sqrt{\epsilon+m^2}$. However, this can become arbitrarily large for large $m$. Hence, the sequence $s_n$ is not Cauchy.
I figure this can be written more formally, but I just want to know if what I typed is correct.

I'll detail my comment: Let $a_n$ be a cauchy sequence: $ \epsilon >0 ;   m,n>N \implies |a_n-a_m|<\epsilon. $
$ \implies n>N ; |a_n| \leq \epsilon + |a_{N+1}| $ 
Hence $(a_n)$ is bounded.
Now your sequence is obviously not bounded, so it cannot be a cauchy sequence. 
