In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$. Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $p^2<2$, and $B$ consists of all positive rationals $p$ such that $p^2>2$. He shows that $A$ contains no largest number and $B$ contains no smallest.
And then in 1.2, Rudin remarks that what he has done above is to show that the rational number system has certain gaps. His remarks confused me.
My questions are:
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If he had shown that no rational number $p$ with $p^2=2$, this already gave the conclusion that rational number system has "gaps" or "holes". Why did he need to set up the second argument about the two sets $A$ and $B$?
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How does the second argument that "$A$ contains no largest number and $B$ contains no smallest" showed gaps in rational number system? My intuition does not work here. Or it is nothing to do with intuition?
from Hot Weekly Questions - Mathematics Stack Exchange
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