Where does the less than or equal come from in the proof of density of Q in R?

[completenoob]

Hello,

I have decided to study analysis on my own and am starting with principles of mathematical analysis by rudin.
I am having trouble understanding pg. 9 on the density of Q in R, part b.

It states:
If $x \in R, y \in R$ and $x<y$ the there exists a $p \in Q$ such that $x < p < y$
Proof:
Since $x<y$, we have $y-x>0$ and the Archemedian Property furnishes a positive integer n such that:
$n(y-x)>1$
Applying the AP again, to obtain positive integers m1 & m2 such that $m1>nx$, $m2>-nx$
Then: $-m2<nx<m1$
Hence there is an integer m such that $m-1 \le nx<m$...

Can someone explain to me the last line? Where does this less then or equal come from?

 

[completenoob]

 

[JinM]

https://www.physicsforums.com/showthread.php?p=1896145#post1896145

Post 8 onwards.

 

[completenoob]

https://www.physicsforums.com/showthread.php?p=1896145#post1896145

Post 8 onwards.

Thank you JinM, that helped a lot. However, on your last post in that thread, It does not make sense to me that there exists a m-1 when m is the minimum. Wouldn't it make sense to say that m-1 is the minimum?

 

[completenoob]

MMmm...For being my first analysis book Principles of Mathematical Analysis by Rudin might be a little too hard. Should I study Apostol's Principles of Mathematical Analysis first? Then go back to Rudin's text?

 

[uman]

Probably depends how much you already know.

 

[completenoob]

Calc 1-3. No analysis experience.

 

[completenoob]

Yah i need a text for self study. I have no experience with proofs.

 

[slider142]

If you can get your hands on it, Spivak's "Calculus" is a great text for bridging the mechanics of calculus and real analysis, and is a bit easier than Rudin. Also, "Advanced Calculus: A Friendly Approach" is very readable cover to cover with good examples and exercises.