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

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

Let $\sequence {x_n}$ be any ordinal-valued sequence.

Then $\sequence {x_n}$ is bounded above.

Proof

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

Thus by Mapping from Set to Class of All Ordinals is Bounded Above, $\sequence {x_n}$ has an upper bound.

$\blacksquare$