Why "There exists a positive number y such that for every positive number x ,we have y<x" is a FALSE statement.

Question

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.

Answer 1

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 $\frac y2$.

Answer 2

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 $$\frac{1}{x} < \epsilon$$