# Definition:Upper Bound of Sequence/Real

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*This page is about upper bounds of real sequences which are bounded above. For other uses, see Definition:Upper Bound.*

## Definition

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

Let $\sequence {x_n}$ be bounded above by $H \in \R$.

Then $H$ is an **upper bound of $\sequence {x_n}$**.

### Upper Bound of Number

When considering the upper bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself.

Thus the construction:

*The set of numbers which fulfil the propositional function $P \left({n}\right)$ is bounded above with the upper bound $N$*

would be reported as:

*The number $n$ such that $P \left({n}\right)$ has the upper bound $N$*.

This construct obscures the details of what is actually being stated. Its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is considered an abuse of notation and so discouraged.

This also applies in the case where it is the upper bound of a mapping which is under discussion.

## Also see

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**bounded sequence** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**bounded sequence**