# Limit Ordinal Equals its Union

## Theorem

Let $\lambda$ be a limit ordinal.

Then:

$\lambda = \bigcup \lambda$

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

## Proof

$\bigcup \lambda \subseteq \lambda$

Suppose $x \in \lambda$.

$x^+ < \lambda$

and so:

$x \in x^+$ and $x^+ \in \lambda$

from which:

$x \in \bigcup \lambda$

That is:

$\lambda \subseteq \bigcup \lambda$

Hence by set equality:

$\lambda = \bigcup \lambda$

$\blacksquare$

## Also presented as

This can also be presented in the form:

$\lambda = \ds \bigcup_{\alpha \mathop \in \lambda} \alpha$

which by definition of union can be seen to be equivalent to $\lambda = \bigcup \lambda$.