Real Subtraction is Closed

Theorem

The set of real numbers is closed under subtraction:

$\forall a, b \in \R: a - b \in \R$

Proof

From the definition of real subtraction:

$a - b := a + \paren {-b}$

where $-b$ is the inverse for real number addition.

From Real Numbers under Addition form Group:

$\forall a, b \in \R: a + \paren {-b} \in \R$

Therefore real number subtraction is closed.

$\blacksquare$

Sources

• 1957: Tom M. Apostol: Mathematical Analysis ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-2}$ Arithmetical properties of real numbers: Axiom $4$
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$