Cauchy's Convergence Criterion/Real Numbers

Theorem

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

Then $\sequence {x_n}$ is a Cauchy sequence if and only if $\sequence {x_n}$ is convergent.

Proof

Necessary Condition

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

Let $\sequence {x_n}$ be convergent.

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

$\Box$

Sufficient Condition

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

Let $\sequence {x_n}$ be a Cauchy sequence.

Then $\sequence {x_n}$ is convergent.

$\Box$

The conditions are shown to be equivalent.

Hence the result.

$\blacksquare$

Also known as

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

Also see

• Real Number Line is Banach Space
• Cauchy's Convergence Criterion on Complex Numbers

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Theorem $1.2.9$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Cauchy sequence