## 502 – Infinite well-ordered sets

I just want to record the Exercise I mentioned in class:

Suppose that $(A,<)$ is an infinite well-ordered set, and that $B\subseteq A.$ Show that there is a bijection between $A$ and the disjoint union $A\sqcup B.$

To be explicit, I want a proof that makes no use of the axiom of choice. Also, although I am not requiring this as an exercise, recall that the point is to use this result to complete the proof of the following:

Theorem. If $(A,<)$ is a well-ordered set, then there is a well-ordered set $(B,\prec)$ such that $(A,<)$ is a proper initial segment of $(B,\prec)$ and there is no injection from $B$ into $A.$