# Multiplication of Positive Number by Real Number Greater than One

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## 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 |

$\blacksquare$