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

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

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

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.
$\blacksquare$