Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 1

Proof
Let $\left \langle {a_n} \right \rangle$ be a Cauchy sequence in $\R$.

We have the result Real Number Line is Metric Space.

Hence by Convergent Subsequence of Cauchy Sequence in Metric Space, it is sufficient to show that $\left \langle {a_n} \right \rangle$ has a convergent subsequence.

Since $\left \langle {a_n} \right \rangle$ is Cauchy, by Real Cauchy Sequence is Bounded, it is also bounded.

By the Bolzano-Weierstrass Theorem, $\left \langle {a_n} \right \rangle$ has a convergent subsequence.

Hence the result.