# Proving that an initial infinite ordinal is a limit ordinal

As in the title,

An initial infinite ordinal is a limit ordinal.

How can I start proving that?

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Start by writing down what "initial" and "limit" mean. –  Chris Eagle Jan 7 '13 at 0:21

Hint: Show that there is a bijection between $\alpha$ and $\alpha+1$ for infinite ordinals $\alpha$.
Consider the case between $\omega$ and $\omega+1$, and see that it transfers to the other cases easily.