## Proof That Taking The Reciprocal Reverses the Direction of the Inequality For Positive Numbers

If
$x \gt y \gt 0$
is
$\frac{1}{y} \gt \frac{1}{x} \gt 0$
?
Yes it is.
To prove it divide by
$x$
and
$y$

$x \gt y \gt 0$

$\frac{x}{x} \gt \frac{y}{x} \gt 0$

$1 \gt \frac{y}{x} \gt 0$

$\frac{1}{y} \gt \frac{1}{x} \gt 0$

This is only the case for
$x \gt y \gt 0$
since multiplying or dividing by a negative number changes the direction of the inequality.