I've come into trouble while trying to prove that $\sup\left\{n^2 + n + 1 \mid n \in \mathbb{N} \right\} = +\infty.$
While first statement of supremum is apparent i.e. $(\forall n \in \mathbb{N})(n^2 + n + 1 < +\infty)$, I can't come up with idea, how to prove the second one $(\forall \alpha \in \mathbb{R}, \alpha < +\infty)(\exists n \in \mathbb{N})(n^2 + n + 1 > \alpha).$
My try is based on fact, that because there is exists quantifier, I can chose some fixed $n$ which will then accomplish $n^2 + n + 1 > \alpha$. So if someone gives me $\alpha < +\infty$ which is a fixed number, I'm taking $n = \left \lfloor{\alpha}\right \rfloor + 1$.
Is my approach correct?
Hint
$$n^2+n+1\geq n$$ for all $n\geq 1$.