Proving that there is no negative integer with $n^2+n<0$ I'm trying to prove the statement: There is no negative integer with $n^2+n<0$. For this, I went with proving it by contradiction. Here's what I have so far:
Let us assume there is a negative integer $n$ with $n^2+n<0$. Since $n$ is negative, this means that $n<0$. Then, $n^2\ge 0$, where $n^2=n*n$. However, this is a contradiction because $n^2\ge 0$ and we assumed $n^2+n<0$, $n^2+n$ will at least be $0$ and $0 \nless 0$.
I'm not sure if my proof is correct or even going in the right direction, I think my negated statement may be wrong, which makes my proof wrong too. Any feedback or help is appreciated.

In your argument it is not clear how you conclude that $n^{2}+n $ is at least $0$.
Let $m =-n$. Them $m >0$ and $n^{2}+n <0$ gives $m^{2} <m$. But then   you can cancel $m$ and get $m <1$. there is no positive integer $m$ such that $m <1$ so we have a contradiction. 
