Real Number Greater than One is Less than Square

Theorem

Let $x \in \R$.

Let $x > 1$.

Then:

$x^2 > x$

Proof

As $x > 1$ it follows that $x > 0$.

Thus by Real Number Ordering is Compatible with Multiplication:

$x \times x > 1 \times x$

and the result follows.

$\blacksquare$

Sources

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