$-\mathbb{N}$ is not bounded below, right?

Parable:
I am asked to prove by contradiction that the set of negative integers $\mathbb{N}$ is not bounded below. My professor writes
$$-\inf(-\mathbb{N})=\sup(\mathbb{N}),$$
and says that since $LHS\in\mathbb{R}$ and $RHS\not\in\mathbb{R}$, by the Archimedean Property, then by contradiction we have that the set of negative integers $\mathbb{N}$ is not bounded below, as desired.

Question:
Could someone explain to me how this is so? I do not yet see how it is so.

The equality $\sup (S)=-\inf (-S)$ holds for any set of reals $S$. It implies that if $l$ is a lower bound for $-\Bbb N$, then $-l$ is an upper bound for $\Bbb N$. But $\Bbb N$ has no upper bound, because by the Archimedean Property, for every $x\in \Bbb R$ there is an $n\in \Bbb N$ with $x\lt n$. Therefore $-\Bbb N$ cannot have a lower bound.