How do I apply the fact that every nonempty subset of a well-ordered set has the smallest element in this proof?

Question

I have to prove this: Show that in a well-ordered set, every element except the largest(if one exists) has an immediate successor.

Here is what I have: Let A be a well-ordered set.

let a belong to the set A.

Consider the set S = {b belongs to set A |a < b}

We want to use the fact that every nonempty subset of a well-ordered set has the smallest element.

If a is not the largest element of the set, then (a,b) = {x | a < x < b) = nonempty set, and has the smallest element b. Then b is the immediate successor of a if a<b and if there is no x such that a < x < b.

Therefore, in a well-ordered set, every element except the largest (if one exists) has an immediate successor.

I am correct?

Answer 1

No, that does not work: you pulled $b$ out of thin air. At the very least you need to say that if $a$ is not the largest element of $A$, then there is some $b\in A$ such that $a<b$; then you can consider the interval $(a,b)$. However, that interval may be empty, in which case it does not have a smallest element. And even if it is non-empty and therefore does have one, you cannot call that element $b$: you’ve already used that name for something else.

But in fact there’s no reason to look at such an interval in the first place. The point is that if $a$ is not the largest element of $A$, then $S\ne\varnothing$, and therefore $S$ has a least element $s$. Now show that $s$ is an immediate successor of $a$ by showing that there is no $b$ such that $a<b<s$.

Answer 2

Suppose that $a$ is not the largest element. This means by definition that $U_a:=\{x: x > a\}$ is non-empty. As $X$ is a well order $\min(U_a)$ exists.

As $\min(U_a) \in U_a$ we know $a < \min(U_a)$. If there would exist some $b \in (a, U_a)$, then $b > a$ so $b \in U_a$ but at the same time $b < \min(U_a)$ contradicting the minimality of $\min(U_a)$. So no such $b$ can exist and so $(a, \min(U_a))$ is empty.

This is by definition what it means for $\min(U_a)$ to be the immediate successor of $a$, often denoted by $a^+$ in order theory.