Cauchy's Convergence Criterion/Real Numbers

From ProofWiki
Jump to navigation Jump to search

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


Source of Name

This entry was named for Augustin Louis Cauchy.


Sources