Real Number Line is Banach Space

Theorem

The real number line, along with the absolute value function, forms a Banach space over $\R$.

Proof

From Real Numbers form Vector Space we have the real number line $\R$ is a vector space over $\R$.

That the norm axioms are satisfied by the absolute value function is proven in Absolute Value is Norm.

Then we have Real Number Line is Complete Metric Space.

Hence the result.

$\blacksquare$