# Cauchy's Convergence Criterion/Real Numbers/Necessary Condition/Proof 1

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## Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.

Let $\sequence {x_n}$ be convergent.

Then $\sequence {x_n}$ is a Cauchy sequence.

## Proof

Let $\sequence {x_n}$ be convergent.

Let $\struct {\R, d}$ be the metric space formed from $\R$ and the usual (Euclidean) metric:

- $\map d {x_1, x_2} = \size {x_1 - x_2}$

where $\size x$ is the absolute value of $x$.

This is proven to be a metric space in Real Number Line is Metric Space.

From Convergent Sequence in Metric Space is Cauchy Sequence, we have that every convergent sequence in a metric space is a Cauchy sequence.

Hence $\sequence {x_n}$ is a Cauchy sequence.

$\blacksquare$

## Also known as

**Cauchy's Convergence Criterion** is also known as the **Cauchy convergence condition**.

## Sources

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- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Theorem $1.2.9$