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 the definition of sequence, the domain of $\left\langle{x_n}\right\rangle$ is a subset of $\omega$, the first infinite ordinal.

Let $f$ be the mapping extending $\left\langle{x_n}\right\rangle$ to $\omega$ by letting $f(k) = x_k$ when $k \in \operatorname{dom} \left\langle{x_n}\right\rangle$ and $f(k) = 0$.

Then by Mapping from Ordinal to Ordinal Class is Bounded Above, $f$ has an upper bound, which is then an upper bound of the sequence.