Absolute Value is Norm

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Theorem

The absolute value is a norm on the set of real numbers $\R$.


Proof

By Complex Modulus is Norm then the complex modulus satisfies the norm axioms on the set of complex numbers $\C$.

Since the real numbers $\R$ is a subset of the complex numbers $\C$ then the complex modulus satisfies the norm axioms on the real numbers $\R$.

By Complex Modulus of Real Number equals Absolute Value then the absolute value satisfies the norm axioms on set of real numbers $\R$.

$\blacksquare$