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## Main Question or Discussion Point

I am reading Rudin's proof of this property, but I find one assertion he makes quite disagreeable to my understanding; I am hoping that someone could expound on this assertion. Here is the statement and proof of the archimedean property:

(a) If ##x \in R##, ##y \in R##, and ##x > 0##, then there exists a positive integer ##n## such that ##nx > y##.

(a) Let A be the set of all nx. If (a) were false, then y would be an upper bound of A.

I understand the method of proof he is employing, a proof by contradiction, which leads to y being larger than every element A (hence, it is an upper bound). However, I do not understand how this immediately implies that A

EDIT: Also, what is the quantification on x and y?

(a) If ##x \in R##, ##y \in R##, and ##x > 0##, then there exists a positive integer ##n## such that ##nx > y##.

(a) Let A be the set of all nx. If (a) were false, then y would be an upper bound of A.

**But then A has a least upper bound in R**...I understand the method of proof he is employing, a proof by contradiction, which leads to y being larger than every element A (hence, it is an upper bound). However, I do not understand how this immediately implies that A

*must*have a least upper bound. Which theorem is he invoking to come to such a conclusion?EDIT: Also, what is the quantification on x and y?