# Definition:Upper Bound of Sequence/Real

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.