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.

For $n \neq m$, $|n^{2} - m^{2}| \in \mathbb{Z}$, $|n^{2} - m^{2}| \neq 0$. There isn't an integer between $0$ and $1$, so if you pick $\epsilon = \frac{1}{2}$, then $|m^{2} - n^{2}| > \epsilon$ whenever $m \neq n$.
