# Definition:Bounded Sequence/Real

This page is about Bounded Real Sequence. For other uses, see Bounded.

## Definition

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

Then $\sequence {x_n}$ is bounded if and only if $\exists m, M \in \R$ such that $\forall i \in \N$:

$m \le x_i$
$x_i \le M$

That is, if and only if it is bounded above and bounded below.

## Unbounded

$\sequence {x_n}$ is unbounded if and only if it is not bounded.