Real Number Line is Complete Metric Space

Theorem
The real numbers $$\R$$, equipped with the usual Euclidean metric, are a complete metric space.

Proof
See Real Number Line is Metric Space for the proof that $$d(x,y):=|x-y|$$ is a metric.

It remains to show that this space is complete; i.e. that every Cauchy sequence of real numbers has a limit. So let $$(a_n)$$ be a Cauchy sequence. It is sufficient to show that $(a_n)$ has a convergent subsequence.

We observe that the fact that $$(a_n)$$ is Cauchy implies that $$(a_n)$$ is bounded. Indeed, there exists $$n_0\in\N$$ such that
 * $$ |a_m - a_n| < 1 $$

for all $$m,n\geq n_0$$. In particular,
 * $$ |a_m| = |a_n + a_m - a_n| \leq |a_n| + |a_m - a_n| \leq |a_n|+1$$

for all $$m\geq n_0$$, so the sequence is bounded as claimed.

By the Bolzano-Weierstrass Theorem, $$(a_n)$$ has a convergent subsequence, and we are done.