If every subset of a total order $(P,\leq)$ is isomorphic to an initial segment of $(P,\leq)$, then is $P$ a well-order?

Question

I have to proof that "If every subset of a total order $(P,\leq)$ is isomorphic to an initial segment of $(P,\leq)$, then $P$ is a well-order". I already got the proof of the converse and this direction seems much easier to me, but I can't see it. Am I missing something obvious?

Answer 1

Suppose that $\langle P,\le\rangle$ is not a well-order. Then there is a non-empty $A\subseteq P$ with no least element. $A$ is isomorphic to an initial segment of $P$, so $P$ has no least element. But every subset of $P$ is isomorphic to an initial segment of $P$, so no subset of $P$ has a least element. Is that possible?