# Cauchy Sequence Converges on Real Number Line/Proof 1

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## Theorem

Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.

Then $\sequence {a_n}$ is convergent.

In other words, $\struct {\R, \size {\,\cdot\,}}$ is a Banach space.

## Proof

Let $\sequence {a_n}$ 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 $\sequence {a_n}$ has a convergent subsequence.

Since $\sequence {a_n}$ is Cauchy, by Real Cauchy Sequence is Bounded, it is also bounded.

By the Bolzano-Weierstrass Theorem, $\sequence {a_n}$ has a convergent subsequence.

Hence the result.

$\blacksquare$