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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.

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