Proof That Taking The Reciprocal Reverses the Direction of the Inequality For Positive Numbers
\[x \gt y \gt 0\]
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. 
