Definition:Bounded Sequence/Real

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This page is about real sequences which are bounded. For other uses, see Definition: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

$\left \langle {x_n} \right \rangle$ is unbounded if and only if it is not bounded.


Sources