# Real Number between Zero and One is Greater than Square

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## 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.

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.8 \ (1) \ \text{(i)}$