Strictly Positive Real Numbers are not Closed under Subtraction

Theorem

The set $\R_{>0}$ of strictly positive real numbers is not closed under subtraction.

Proof

Proof by Counterexample

Let $a = 1$ and $b = 2$.

Then:

$a - b = -1$

but $-1$ is not a (strictly) positive real number.

$\blacksquare$

Sources

• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$