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. 

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