The density proof is that for every $x,y \in \mathbb{R}, \exists r \in \mathbb{Q} \ s.t \ x < r < y$
My professor gave an alternative proof to the one in Rudin. It starts by saying:
By Archimedean property, $ \exists n \in \mathbb{N} \ s.t. \ > n(y-x)>1$. Let $E=\{ m \in \mathbb{Z} : m < n \cdot y \} $.
(We want to show that E has a maximal element $m_0$ such that it $nx < m_0 < ny$ so that $\frac{m_0}{n}$ will be the fraction we want.)
Claim: E has a maximal element. By a theorem, E will contain a Maximal if $E \neq \emptyset$ and if E is bounded above. By construction, $n \cdot y$ is an upper bound for E.
Here's the part I don't understand:
By Archimedean property, $\exists m \in \mathbb{N}$ such that $ -m>-ny$ and $-m \in E$ so $E \neq \emptyset$.
I don't understand why we started taking negatives of the statement $ m < n \cdot y$. And how could $-m \in E$? I'm sorry if this is a really simple question, but this has been driving me crazy.