bedi said:
Let X be an initial segment of a set A. By definition, if x is in X, a is in A and x>a then a is in X too. Can we say that some elements of A that are greater than x are also in X? Or X only consists of elements smaller than x?
No, that's why the word "initial" is used. If X is an initial segment of A and x is a specific member, then that definition says that all members of A less than x are in X. It does NOT say anything about numbers larger than x, one way or the other.
Suppose A is the set of all positive integers less than 10. Let X be the set {1, 2, 3, 4, 5}. Do you see why that is an "initial segment" of A?
If, for example, x= 3, then both 1 and 2, all members of A less than 3 are in the set. As for members of X larger than 3, some, 4 and 5, are in the set, some, 6, 7, 8, and 9, are not.
What about {2, 3, 4} or {5, 6, 7, 8, 9}