I am doing the prof for the denseness of $\Bbb Q$. I want to find $m,n$ in $\Bbb Z$ ($n>0$) such that $a<\frac mn<b$.
My prof says the first step is to find $n$ in $\Bbb N$ such that $bn-an>1$.
And the second step is to prove the following claim: if [left with] two real numbers $r_1,r_2 \in \Bbb R$ [such that] $r_2-r_1>1$, then there’s always an integer between, so there exists an $m$ in $\Bbb Z$ such that $r_1<m<r_2$.
To be honest, I haven’t even got to why the second step is the way it is, however I am confused when it comes to the first step; it’s like the inequality $bn-an>1$ just pops out of nowhere. I’d appreciate some help. Please and thank you.
You have$$bn-an>1\iff(b-a)n>1\iff n>\frac1{b-a}.$$And there is such a $n$ by the Archimedian property.