A linear order that every proper initial segment is $\operatorname{seg}(x)$ for some $x$ is a well-order

Question

Can someone help with the following problem: Let $\langle A, R\rangle$ is a linear ordered set. Let every time when $w$ is an initial segment of $\langle A, R\rangle$, to be true that $w=A$ or $(∃x ∈ A)(w = \operatorname{seg}(x))$. Prove that $\langle A,R \rangle$ is a well ordered set.

Any help is appreciated, thanks in advance!

Answer 1

Given a nonempty subset $S$ of $A$, Apply the given assumption to the set $w$ of all strict lower bounds of $S$.