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.

$$n(n+1)<0$$
If $n>0,n+1<0\iff n<-1$ which is untenable
If $n<0,n+1>0\iff n>-1\iff -1<n<0$
