Real Number between Zero and One is Greater than Square
Let $x \in \R$.
Let $0 < x < 1$.
- $0 < x^2 < x$
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.
We have that Real Numbers form Ordered Integral Domain.
Thus Square of Element Less than Unity in Ordered Integral Domain applies directly.