I am trying to self-study Baby Rudin (and it's proving quite challenging to me)
Could someone clarify where the underlined part comes from?
Text:
(a) If $x \in R, y \in R,$ and $x > 0$, then there is a positive integer $n$ such that $nx > y$.
Proof (a) Let $A$ be the set of all $nx$, where $n$ runs through the positive integers. If (a) were false, then $y$ would be an upper bound of $A$. But then $A$ has a least upper bound in $\mathbb{R}$. Put $\alpha = \sup A$. Since $x > 0$, $\alpha - x < \alpha$, and $\alpha - x$ is not an upper bound of $A$. $\underline{\text{Hence $\alpha - x < mx$ for some positive integer $m$}}$. But then $\alpha < (m+1)x \in A$, which is impossible, since $\alpha$ is an upper bound of $A$.
Thanks in advance