Mapping from Set to Class of All Ordinals is Bounded Above/Sequence Corollary

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

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

Proof
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, $\left\langle{}\right\rangle$ has an upper bound.