Real Number Line is Complete Metric Space

Theorem

The real number line $\R$ with the usual (Euclidean) metric forms a complete metric space.

Proof

From Real Number Line is Metric Space, the distance function defined as $\map d {x, y} = \size {x - y}$ is a metric on $\R$.

It remains to be shown that the metric space $\struct {\R, d}$ is complete.

By definition, this is done by demonstrating that every Cauchy sequence of real numbers has a limit.

This is demonstrated in Cauchy's Convergence Criterion.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $1$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): metric space
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): metric space