Cauchy's Convergence Criterion

Theorem

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is convergent iff $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.

Proof

Sufficient Condition

Suppose $\left \langle {x_n} \right \rangle$ is convergent.

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

We also have that the real number line is a metric space.

Hence $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.

$\Box$

Necessary Condition

Suppose $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.

We have the result that a Cauchy Sequence Converges on Real Number Line.

Hence $\left \langle {x_n} \right \rangle$ is convergent.

$\Box$

The conditions have been shown to be equivalent.

Hence the result.

$\blacksquare$

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Theorem $1.2.9$