Length of Union of Chain of Ordinal Sequences

Theorem
Let $C$ be a chain of ordinal sequences.

Let $\lambda$ be a limit ordinal such that all elements of $C$ are of length (strictly) less than $\lambda$.

Suppose that for every ordinal $\alpha < \lambda$ there exists $\theta \in C$ such that:
 * $\size \theta = \alpha$

where $\size \theta$ denotes the length of $\theta$.

Then:
 * $\size {\bigcup C} = \lambda$

where $\bigcup C$ denotes the union of $C$.