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denotes the distance between and .

Applications and important results

If and are convergent sequences, then the following limits exist, and can be computed as follows:

*
* for all real numbers
*
* , provided that
* for all and

Moreover:
* If for all greater than some , then .
* (Squeeze theorem)
If is a sequence such that for all and $\backslash lim\_\{n\backslash to\backslash inftya\_n\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; b\_n\; =\; L$,}}
then is convergent, and .
* If a sequence is bounded and monotonic then it is convergent.
* A sequence is convergent if and only if all of its subsequences are convergent.

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