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

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 $1.2.9$