# Multiplication of Positive Number by Real Number Greater than One

## Theorem

Let $x$ and $y$ be real numbers.

Let $x > 1$.

Let $y > 0$.

Then $\dfrac y x < y$.

## Proof

 $\displaystyle x$ $<$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 x$ $<$ $\displaystyle 1$ Ordering of Reciprocals $\displaystyle \leadsto \ \$ $\displaystyle \frac y x$ $<$ $\displaystyle y$ Real Number Ordering is Compatible with Multiplication

$\blacksquare$