# Real Number between Zero and One is Greater than Square

## Theorem

Let $x \in \R$.

Let $0 < x < 1$.

Then:

$0 < x^2 < x$

## Proof 1

We are given that $0 < x < 1$.

By direct application of Real Number Ordering is Compatible with Multiplication, it follows that:

$0 \times x < x \times x < 1 \times x$

and the result follows.

$\blacksquare$

## Proof 2

We have that Real Numbers form Ordered Integral Domain.

Thus Square of Element Less than Unity in Ordered Integral Domain applies directly.

$\blacksquare$