I believe my question is different to this one.
In Rudin Principle's of Mathematical Analysis Example $1.1$, Rudin first shows that $p^2=2$ is not satisfied by any rational $p.$ Then he defines $A$ to be the set of all positive rationals $p$ such that $p^2 < 2, $ and shows that $A$ contains no largest member. This is before the definition of $\sup$ has been introduced.
Rudin's idea of the above discussion was to show that, "the rational number system has certain gaps... and the real number system fills these gaps." My problem with this is that I don't think Rudin has shown that the rational number system has gaps, like he claims. If we instead define $A$ to be the set of positive real numbers such that $p^2 < 2,$ then everything in the above discussion still holds true. By Rudin's logic, we could conclude that "the real number system has certain gaps", which is not true. His conclusion makes more sense if instead he defined $A$ to be the set of all positive rationals such that $p^2 \leq 2.$
Is my above analysis correct? Should Rudin have started with, "Define $A$ to be the set of all positive rationals $p$ such that $p^2 \leq 2."$
Attempt at answering my own question:
I am correct, but also, we have just established that $p^2=2$ is not satisfied by any rational $p,$ and therefore the set of all positive rationals $p$ such that $p^2 \leq 2 $ is equivalent to the set of all positive rationals $p$ such that $p^2 < 2.$ So the $<$ and $\leq$ are interchangeable here, so it doesn't make a difference which one we use.
Thoughts?
