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For me, it seems both the above mentioned statements are true and are equivalent. But, in my textbook, it is written otherwise. Please let me know if textbook has a printing mistake or where i am getting wrong. Thanks.

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    The text is correct. Note that $x$ and $y$ appear in a different order in the first phrase of the second sentence than they do in the first. – saulspatz Aug 09 '20 at 14:47
  • But at the end, both seems to imply the same meaning. And can you explain why (2) is false?? –  Aug 09 '20 at 14:49
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    But the burden and import of the text’s paragraph is that the quantifiers and their order is of overwhelming importance. (Don’t look merely at the algebraic sentence, see how it is introduced.) – Lubin Aug 09 '20 at 14:56
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    Think of it as a game in which we successively play positive reals and you win if your number $y$ is smaller than my number $x$. (1) You can be sure of winning if I have to choose my $x$ first. (2) You can't be sure of winning if you must choose your $y$ first. – Ned Aug 09 '20 at 15:08
  • Great explanation Ned. Thanks!! –  Aug 09 '20 at 15:09

2 Answers2

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The first sentence says, for every positive number $x$ you name, I can name a smaller positive number $y$. Sure, I just take $y=\frac x2$,

The second sentence says there's a positive number $y$ that's smaller than any positive number $x$ you can name. How does that work? I name $y$, you name $\frac y2$.

saulspatz
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The new situation where this can be accomplished is the ordered field of rationa functions with real coefficients. For polynomials $p(x), q(x)$ where $q$ is not allowed to be the constant $0$ polynomial, although it is permitted to have (a finite number of) roots, we call $\frac{p}{q}$ "positive" if there is a (large) positive number $X$ such that for every $x > X,$ we have $\frac{p(x)}{q(x)} > 0.$ So, for instance, $$ \frac{x - 1234567890}{x^{57}} $$ is a positive element in this ordered field.

Next, given two rational functions, say $r(x)$ and $s(x),$ we say that $r > s$ if $r-s$ is "positive," there is a $X$ such that every $x > X$ gives $r(x) - s(x)$ positive.

This field does have a copy of the real numbers (as constant polynomials/rational functions). It also has infinitesimal elements, gy our definitions $\frac{1}{x}$ is infinitesimal, in tha, for any real positive constant $\epsilon,$ we have $\epsilon$ greater than the function $\frac{1}{x}.$ Indeed, we can take $$ X = \frac{1}{\epsilon}. $$ Whenever $$ x > X = \frac{1}{\epsilon} $$ we get $$\frac{1}{x} < \epsilon$$

Will Jagy
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