Mapping from Set to Ordinal Class is Bounded Above/Sequence Corollary

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Corollary to Mapping from Set to Ordinal Class is Bounded Above

Let $\left\langle{x_n}\right\rangle$ be any ordinal-valued sequence.


Then $\left\langle{x_n}\right\rangle$ is bounded above.


Proof

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By definition, a sequence $\left\langle{x_n}\right\rangle$ is a mapping whose domain is a subset of the natural numbers and is thus a set.

Thus by Mapping from Set to Ordinal Class is Bounded Above, $\left\langle{x_n}\right\rangle$ has an upper bound.

$\blacksquare$