I've recently been delving into following the proof of 'AC implies Zorn's lemma' by Lewin. I spent a significant amount of time understanding the meaning of an initial segment and how it is handled in the proof. As a beginner to the elementary set theory, it was really hard for me though many of you might regard it the easiest one.
The initial segment has several definitions and one of them is: For a totally ordered set $(A, \le)$ and $c \in A$ $$ P(A,c) = \{x \in A~|~x < c\} $$ is called an initial segment in $A$.
But I am not still confident whether I can utilize the notion of the initial segment in the proof of any other proposition. Could you give some examples where it can be meaningfully used? Please comment about any insight I should catch from them.